Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling

TL;DR

This paper analyzes mathematical models of cell movement influenced by chemical signals, focusing on simplifications in different dimensions and their convergence rates.

Contribution

The paper introduces new methods for analyzing singular limits in chemotaxis systems up to critical dimensions.

Findings

Parabolic-elliptic simplification is studied up to the critical dimension N=4.

Indirect-direct simplification is analyzed up to the critical dimension N=2.

Convergence rates and initial layer effects are revealed for both scenarios.

Abstract

Singular limits for the following indirect signalling chemotaxis system \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t n = \Delta n - \nabla \cdot (n \nabla c ) & \text {in } \Omega \times (0,\infty ) , \\ \varepsilon \partial _t c = \Delta c - c + w & \text {in } \Omega \times (0,\infty ), \\ \varepsilon \partial _t w = \tau \Delta w - w + n & \text {in } \Omega \times (0,\infty ), \\ \partial _\nu n = \partial _\nu c = \partial _\nu w = 0, & \text {on } \partial \Omega \times (0,\infty ) \end{array} \right. \end{aligned}$$\end{document}∂tn=Δn-∇·(n∇c)inΩ×(0,∞),ε∂tc=Δc-c+winΩ×(0,∞),ε∂tw=τΔw-w+ninΩ×(0,∞),∂νn=∂νc=∂νw=0,on∂Ω×(0,∞)are investigated. More precisely, we study parabolic-elliptic simplification, or PES, \documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon \rightarrow 0^+$$\end{document}ε→0+ with fixed \documentclass[12pt]{minimal}
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\begin{document}$$\tau >0$$\end{document}τ>0 up to the critical dimension \documentclass[12pt]{minimal}
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\begin{document}$$N=4$$\end{document}N=4, and indirect-direct simplification, or IDS, \documentclass[12pt]{minimal}
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\begin{document}$$(\varepsilon ,\tau )\rightarrow (0^+,0^+)$$\end{document}(ε,τ)→(0+,0+) up to the critical dimension \documentclass[12pt]{minimal}
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\begin{document}$$N=2$$\end{document}N=2. These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the \documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}Lp-energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.

Full text

Introduction

The term chemotaxis has been widely used to describe the directed movement of a species responding to a stimulus, with numerous applications in bacterial aggregation [4, 8], cell invasion [5, 38], food chains [40, 44], and other contexts. In mathematical modelling, it turns into cross-diffusive terms in parabolic-parabolic or parabolic-elliptic systems of PDEs. Recently, chemotaxis systems with indirect signalling mechanisms have gained a lot of attention, where a system may include one species and two signals, or two species and one signal. Besides the suggestion of better responses of a species to the environment, see e.g. [31], the differences between the direct and indirect signalling also raise many interesting analytical questions, regarding the global solvability and uniform boundedness [11, 39], infinite-time aggregation [43, 45], large-time behaviours [22, 48], or singular limits [23, 25].

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^N$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 4$$\end{document} , be a bounded domain with sufficiently smooth boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma := \partial \Omega $$\end{document} . In this work, we study the singular limits \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0^+$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varepsilon ,\tau ) \rightarrow (0^+, 0^+)$$\end{document} of the following indirect signalling chemotaxis system

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t n & \hspace{-0.3cm}=& \hspace{-0.2cm} \Delta n - \nabla \cdot (n \nabla c) & \text {in } \Omega \times (0,\infty ) , \\ \varepsilon \partial _t c & \hspace{-0.3cm}=& \hspace{-0.2cm} \Delta c - c + w & \text {in } \Omega \times (0,\infty ), \\ \varepsilon \partial _t w & \hspace{-0.3cm}=& \hspace{-0.2cm} \tau \Delta w - w + n & \text {in } \Omega \times (0,\infty ), \end{array} \right. \end{aligned}$$\end{document}

which is subjected to the no-flux boundary conditions

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial n}{\partial \nu } = \frac{\partial c}{\partial \nu } = \frac{\partial w}{\partial \nu }= 0 \quad \text {on } \Gamma \times (0,\infty ) , \end{aligned}$$\end{document}

and the initial condition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (n,c,w)|_{t=0}=(n_{0},c_{0},w_{0}) \quad \text {on } \Omega , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{0},c_{0},w_{0}$$\end{document} are given smooth data. This system has been studied in [25, 41] to model the movement of Mountain Pine Beetles in a forest habitat \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =0$$\end{document} , where n and w represent the densities of the flying and nesting species, and c is the concentration of beetle pheromones. In [11], the authors studied System (1.1), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =\tau =1$$\end{document} , which models the aggregation phenomena of microglia cells in the Alzheimer disease, where n represents a species density and c, w are the concentrations of two different chemicals. A variant of (1.1) with the setting in the whole spatial domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^4$$\end{document} can be found in [14]. For related models concerning indirect signalling, we refer the reader to [22, 23, 39, 43, 45, 48] and references therein.

Biologically, signals can diffuse on a much faster time scale than the species self-diffusion, which leads to mathematical models that include a sufficiently small parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\varepsilon \ll 1$$\end{document} appearing in front of the time evolution of the signal concentration (i.e., its time derivatives). This scenario has been discussed for the last several decades, where parabolic-parabolic chemotaxis systems had been simplified to their parabolic-elliptic relatives [7, 18]. This type of simplification is well-known as the notion of fast signal diffusion limits or parabolic-elliptic simplification (PES for short) [40, 47], which offers significant benefits not only in mathematical analysis but also in computational simulations. A PES is formally achieved by removing the signal evolution from the considered chemotaxis models, or equivalently, by formally assigning \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon = 0$$\end{document} , leading to an elliptic instead of a parabolic equation for the chemical/signal concentration. However, rigorous analysis of PES has only been conducted in recent works, such as [9, 28, 29, 33, 40, 47]. On the other hand, by setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon = \tau = 0$$\end{document} , we see from the third equation of (1.1) that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \equiv w$$\end{document} , i.e. the two signals coincide, and (1.1) is reduced to a chemotaxis system with a direct signal. Thus, the singular limit problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varepsilon ,\tau )\rightarrow (0^+,0^+)$$\end{document} is called indirect-direct simplification (IDS for short), and has also been considered for related problems in e.g. [23, 25].

The main goals of this work are to study PES and IDS for (1.1) up to the critical dimensions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} , respectively, where we prove the convergence and estimate the convergence rates including the initial layer effect. In the following, we first give the state of the art, which helps to highlight the motivation and novelty of our work. Then, we present our main results as well as the key ideas.

State of the art

The study of PES has been initiated in recent years, with the first work focusing on the classical parabolic-parabolic Keller-Segel model

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} \partial _t u_\lambda = \Delta u_\lambda - \chi \nabla \cdot ( u_\lambda \nabla v_\lambda ) & \text {in } \Omega \times (0,\infty ), \\ \lambda \partial _t v_\lambda = \Delta v_\lambda - v_\lambda + u_\lambda & \text {in } \Omega \times (0,\infty ), \\ (u_\lambda ,v_\lambda )|_{t=0}=(u_{0},v_{0}) & \text {on } \Omega , \end{array} \right. \end{aligned}$$\end{document}

(subjected to the no-flux boundary conditions) and its parabolic-elliptic relative

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} \partial _t u = \Delta u - \chi \nabla \cdot (u \nabla v ) & \text {in } \Omega \times (0,\infty ), \\ \Delta v - v + u = 0 & \text {in } \Omega \times (0,\infty ), \\ u|_{t=0}=u_{0} & \text {on } \Omega . \end{array} \right. \end{aligned}$$\end{document}

In [29], the author positively answered the question: Does the solution of (1.4) converge to that of (1.5) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow 0$$\end{document} ? With sufficiently small and regular initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0,v_0$$\end{document} , the author showed for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\lambda \rightarrow u \text { in } C_{\textsf{loc}}(\overline{\Omega }\times [0,\infty ))$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\lambda \rightarrow v \text { in } C_{\textsf{loc}}(\overline{\Omega }\times (0,\infty )) \cap L^2_{\textsf{loc}}((0,\infty );W^{1,2}(\Omega ))$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow 0$$\end{document} , where the limit (u, v) is the classical solution of (1.5). When the chemotactic flux is of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\lambda S(v_\lambda ) \nabla v_\lambda $$\end{document} (instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\lambda \nabla v_\lambda $$\end{document} ), [28] showed that for a sensitivity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\in C^{1+\vartheta }((0,\infty ))$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in (0,1)$$\end{document} , satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le S(v) \le \chi (a+v)^{-k}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\ge 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>1$$\end{document} , the above convergence holds provided \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi <\chi _*$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _*>0$$\end{document} depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k,a,N,u_0,v_0$$\end{document} . In [9], the author investigated PES for (1.4) but with non-degenerate diffusion of porous medium type. For the whole domain setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega = \mathbb {R}^N$$\end{document} , we refer the reader, for instance, to [19, 33]. This PES has also been investigated also in [47] in the context of Keller-Segel-(Navier-)Stokes system

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} \partial _t n_\varepsilon + u_\varepsilon \cdot \nabla n_\varepsilon = \Delta n_\varepsilon - \nabla \cdot ( n_\varepsilon S(x,n_\varepsilon ,c_\varepsilon ) \cdot \nabla c_\varepsilon ) + f(x,n_\varepsilon ,c_\varepsilon ), \\ \varepsilon \partial _t c_\varepsilon + u_\varepsilon \cdot \nabla c_\varepsilon = \Delta c_\varepsilon - c_\varepsilon + n_\varepsilon , \\ \partial _t u_\varepsilon + \kappa (u_\varepsilon \cdot \nabla ) u_\varepsilon = \Delta u_\varepsilon + \nabla P_\varepsilon + n_\varepsilon \nabla \phi , \; \kappa \in \mathbb R, \nabla \cdot u_\varepsilon = 0 , \\ (n_\varepsilon ,c_\varepsilon ,u_\varepsilon )|_{t=0} = (n_0,c_0,u_0), \end{array} \right. \end{aligned}$$\end{document}

subjected \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{\nu }n_\varepsilon = \partial _{\nu }c_\varepsilon = 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon = 0$$\end{document} on the boundary. It was (conditionally) shown therein that this system can be rigorously simplified to its relative

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} \partial _t n + u \cdot \nabla n = \Delta n - \nabla \cdot ( n S(x,n,c) \cdot \nabla c) + f(x,n,c), \\ u \cdot \nabla c = \Delta c - c + n , \\ \partial _t u + \kappa (u \cdot \nabla ) u = \Delta u + \nabla P + n \nabla \phi , \nabla \cdot u = 0 , \\ (n,u)|_{t=0} = (n_0,u_0), \end{array} \right. \end{aligned}$$\end{document}

via the limit as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} , provided the following uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla c_\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon $$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \Big ( \Vert \nabla c_\varepsilon \Vert _{L^p((0,T);L^q(\Omega ))} + \Vert u_\varepsilon \Vert _{L^\infty ((0,T);L^r(\Omega ))} \Big ) <\infty , \end{aligned}$$\end{document}

for some p, q, r such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<p\le \infty $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>N$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>\max \{2;N\}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{p} + \frac{N}{2q} < \frac{1}{2}$$\end{document} . Related results can be found in [24, 27, 46].

Besides PES, the investigation of IDS has also attracted considerable attention recently. A first work in this direction seems to be [36], where the authors considered a phenotype-switching chemotaxis model, which represents an indirect signalling scheme, of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_\gamma = \Delta u_\gamma - \nabla \cdot (u_\gamma \nabla v_\gamma ) - \gamma u_\gamma + \gamma w_\gamma , & x\in \Omega ,\\ \partial _t v_\gamma = \Delta v_\gamma - v_\gamma + w_\gamma , & x\in \Omega ,\\ \partial _tw_\gamma = \Delta w_\gamma - \gamma w_\gamma + \gamma u_\gamma , & x\in \Omega ,\\ \partial _{\nu }u_\gamma = \partial _{\nu }v_\gamma = \partial _{\nu }w_\gamma = 0, & x\in \Gamma . \end{array}\right. } \end{aligned}$$\end{document}

As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow \infty $$\end{document} , one expects the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\gamma := u_\gamma + w_\gamma , v_\gamma ) \rightarrow (n, v)$$\end{document} where the latter solves the classical Keller-Segel model with direct signalling

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tn = \Delta n - \frac{\theta }{1+\theta }\nabla \cdot (n\nabla v), & x\in \Omega ,\\ \partial _t v = \Delta v - v + \frac{n}{1 + \theta }, & x\in \Omega ,\\ \partial _\nu n = \partial _\nu v = 0, & x\in \Gamma . \end{array}\right. } \end{aligned}$$\end{document}

This convergence was partially shown in [36], and later fully proved in [23]. A similar problem was considered in [25], where the authors studied the following system

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t n_\varepsilon = \Delta n_\varepsilon - \nabla \cdot (n_\varepsilon \nabla c_\varepsilon ) , \\ \varepsilon _1 \partial _t c_\varepsilon = \Delta c_\varepsilon - c_\varepsilon + w_\varepsilon , \\ \varepsilon _2 \partial _t w_\varepsilon = - w_\varepsilon + n_\varepsilon , \\ (n_\varepsilon ,c_\varepsilon ,w_\varepsilon )|_{t=0}=(n_0,c_0,w_0). \end{array} \right. \end{aligned}$$\end{document}

Under the assumption that the initial mass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _\Omega n_0$$\end{document} is sub-critical, i.e. smaller than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\pi $$\end{document} , this system is shown to converge to either

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t n = \Delta n - \nabla \cdot (n \nabla c ) , \\ \partial _t c = \Delta c - c + w , \\ (n,c)|_{t=0}=(n_0,c_0), \end{array} \right. \quad \text { or } \quad \left\{ \begin{array}{lllllll} \partial _t n = \Delta n - \nabla \cdot (n \nabla c ) , \\ \Delta c - c + w = 0, \\ n|_{t=0}=n_0, \end{array} \right. \end{aligned}$$\end{document}

corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _1 =\varepsilon _2 \rightarrow 0$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _1=1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _2 \rightarrow 0$$\end{document} , respectively.

It’s worthwhile to mention that the modelling and analysis of chemotaxis systems with indirect signalling of the type (1.1), both in the parabolic-parabolic and parabolic-elliptic settings, have been subjected to extensive investigation, see e.g. [1, 10, 21, 41, 49, 50] and references therein. Even the question of global existence can be challenging, especially in the critical dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} , see e.g. [11, 14].

Our current work adequately contributes to this literature by investigating the PES and IDS for chemotaxis systems with indirect signalling (1.1)-(1.3) up to the critical dimensions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} , respectively. Furthermore, we also provide the convergence rates, which have been seemingly completely left out in the literature, and reveal the effect of the initial layer.

Main results, challenges and key ideas

Notations: We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{k,p}$$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p\le \infty $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 0$$\end{document} , the usual Lebesgue and Sobolev spaces. Moreover, a general constant C is used for any positive constant that does not depend on spatial and temporal variables, all the unknowns, as well as the relaxation parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ,\tau $$\end{document} . This general constant can vary from line to line, or even within the same line. In case where a dependence is important, such as the dependence on a terminal time T or the diffusion coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , we will write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_T$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_\tau $$\end{document} , etc. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<T\le \infty $$\end{document} , we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _T:= \Omega \times (0,T)$$\end{document}

To study singular limits for (1.1), we impose the following assumption on initial data throughout this work.

Assumption 1.1

The initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_0,c_0,w_0) \in C^1(\bar{\Omega })\times C^2(\bar{\Omega })^2$$\end{document} is nonnegative and satisfied the compatible condition, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\partial n_0}{\partial \nu } = \frac{\partial c_0}{\partial \nu } = \frac{\partial w_0}{\partial \nu } = 0$$\end{document} on the boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} .

Our first main results are about the PES from (1.1)-(1.3) to (1.7)-(1.8). Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} and denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\varepsilon , c_\varepsilon , w_\varepsilon )$$\end{document} the solution of (1.1) with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} . As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} , we formally expect that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\varepsilon ,c_\varepsilon ,w_\varepsilon ) \rightarrow (n,c,w)$$\end{document} , and the limit vector (n, c, w) solves the system

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t n = \Delta n - \nabla \cdot (n \nabla c) & \text {in } \Omega \times (0,\infty ) , \\ \Delta c - c + w = 0 & \text {in } \Omega \times (0,\infty ), \\ \tau \Delta w - w + n = 0 & \text {in } \Omega \times (0,\infty ), \\ \dfrac{\partial n}{\partial \nu } = \dfrac{\partial c}{\partial \nu } = \dfrac{\partial w}{\partial \nu } = 0 & \text {on } \Gamma \times (0,\infty ), \end{array} \right. \end{aligned}$$\end{document}

equipped with the initial value condition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n|_{t=0}=n_{0} \quad \text {on } \Omega . \end{aligned}$$\end{document}

One of the main challenges when connecting solutions of (1.1)-(1.3) and (1.7)-(1.8) or (1.29) is the different structures between the parabolicity and ellipticity and the initial layer, especially in the critical dimensions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} for PES and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} for IDS, see [32]. First, to pass to the limit in a strong sense, the slow evolution (i.e., the products of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and the time derivatives of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon ,w_\varepsilon $$\end{document} ) make the Aubin-Lions lemma difficult to apply. For example, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} maximal regularity applied to the slow-evolution equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \partial _t u_\varepsilon - d \Delta u_\varepsilon + u_\varepsilon = f(x,t)$$\end{document} , associated with the no-flux boundary condition, reads as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \Big ( \Vert \varepsilon \partial _t u_\varepsilon \Vert _{L ^{p}(\Omega \times (0,T))} + \Vert \Delta u_\varepsilon \Vert _{L ^{p}(\Omega \times (0,T))} \Big ) \le \left( \frac{\varepsilon }{p} \right) ^{\frac{1}{p}} \Vert u_0\Vert _{W^{2,p}(\Omega )} + C_{d,p} \Vert f \Vert _{L^p(\Omega \times (0,T))} , \end{aligned}$$\end{document}

see [40, Lemma 3.4], which do not directly give a uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} boundedness for the time derivative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t u_\varepsilon $$\end{document} . Obtaining strong convergence for the slow evolution is tricky and usually requires considerable effort, see e.g. [47]. Second, for fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} , even the global solvability for the system (1.1)-(1.3) in the critical dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} is difficult, see [11, 21]. Some steps in that proof, involving e.g. the use of the heat semigroup or testing the equations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon ,w_\varepsilon $$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon ,-\Delta c_\varepsilon ,w_\varepsilon ,-\Delta w_\varepsilon $$\end{document} heavily depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , and therefore do not yield the required uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} estimates. For instance, the Duhamel principle for the latter slow-evolution equation, represented via the Neumann heat semigroup, is written as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_\varepsilon (x,t) = e^{\frac{1}{\varepsilon }t(d\Delta -I)} u_\varepsilon (x,0) + \frac{1}{\varepsilon }\int _0^t e^{\frac{1}{\varepsilon }(t-s)(d\Delta -I)} f(x,s)ds, \end{aligned}$$\end{document}

which yields that a uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} estimate can only be obtained if the regularity of f is sufficiently regular, at least essentially bounded in time, which is not the case in our situation. Third, it has been numerically demonstrated in [40] that initial data starting far away from the critical manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_{\textsf{PES}}$$\end{document} (see (1.19)) can lead to a significant loss of simplification accuracy. Hence, to achieve simplification accuracy, an analysis of the initial layer is required.

In order to rigorously justify this simplification, we exploit the multiple time scale Lyapunov function, see Lemma 2.2,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal E(n_\varepsilon ,c_\varepsilon ) := \int _\Omega \left( n_\varepsilon (\log n_\varepsilon - c_\varepsilon ) + \frac{1}{2}|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 + \frac{\tau }{2} |\Delta c_\varepsilon |^2 + \frac{1+\tau }{2} |\nabla c_\varepsilon |^2 + \frac{1}{2} c_\varepsilon ^2 \right) , \end{aligned}$$\end{document}

with its dissipation given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \mathcal D(n_\varepsilon ,c_\varepsilon )&:= {-\frac{d}{dt}\mathcal {E}(n_\varepsilon ,c_\varepsilon )}\\&= \int _\Omega \Big ( n_\varepsilon |\nabla (\log n_\varepsilon - c_\varepsilon )|^2 + \frac{1+\tau }{\varepsilon } |\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 + \frac{2}{\varepsilon } |\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 \Big ). \end{aligned} \end{aligned}$$\end{document}

It is remarked that the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon (\log n_\varepsilon - c_\varepsilon )$$\end{document} in the Lyapunov function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}(n_\varepsilon ,c_\varepsilon )$$\end{document} has no sign and needs to be estimated from below. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 3$$\end{document} , the Sobolev embedding is sufficient to absorb the norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon c_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^1(\Omega ))$$\end{document} into the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);H^2(\Omega ))$$\end{document} -norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal E(n_\varepsilon ,c_\varepsilon )$$\end{document} , cf. Lemma 2.3, and to obtain an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} -estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} . In the critical dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} , the method of using the Adam-type inequality, see [11, Section 7], can be adapted to balance the energy-dissipation equality. Unfortunately, because of the slow evolution, the locally spatial truncation argument in [11, Section 8] does not work to control the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} -energy. We overcome this issue by adapting the idea of combining the Sobolev, Gagliardo-Nirenberg, and Young inequalities in [14, Proof of Theorem 1.2]. Then, some feedback arguments, using the heat semigroup as well as maximal regularity with slow evolution, help us to estimate the slow evolution’s components \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon ,c_\varepsilon $$\end{document} .

The strong convergence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon \rightarrow c$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);H^1(\Omega ))$$\end{document} is challenging, see e.g. [47, Section 5], where this was proved by heavily exploiting the higher regularity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} . In this work, we provide a shortened and more direct proof by employing the argument from (2.25)-(2.29), which is basically based on the so-called energy equation method, see e.g. [3, 15]. This method uses the equation obtained by considering an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} energy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(c_\varepsilon -c)$$\end{document} , instead of the energy inequality, and then shows the convergence in norms before using the uniform convexity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);H^1(\Omega ))$$\end{document} to get the strong convergence.

Theorem 1.1

( PES for (1.1)) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 4$$\end{document} and fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} . Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_0,c_0,w_0)$$\end{document} is complied with Assumption 1.1, and furthermore in the critical dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega = B_R$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R>0$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} M{:= \int _{\Omega } n_0} < 64 \tau \pi ^2 . \end{aligned}$$\end{document}

For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\varepsilon ,c_\varepsilon ,w_\varepsilon )$$\end{document} be the global classical solution to parabolic-parabolic system (1.1)-(1.3), given by Theorem 2.1. Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<T<\infty $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{gathered} \sup _{\varepsilon>0} \Big ( \Vert n_\varepsilon \Vert _{C^{\gamma ,\gamma /2}(\overline{\Omega }\times [0,T])} + \Vert n_\varepsilon \Vert _{L^2((0,T);H^1(\Omega ))} \Big ) \le C_{\tau ,T},\\ \sup _{\varepsilon >0} \Big ( \Vert w_\varepsilon \Vert _{L^\infty ((0,T);W^{1,\infty }(\Omega ))} + \Vert \Delta w_\varepsilon \Vert _{L^p(\Omega _T)} + \Vert c_\varepsilon \Vert _{L^\infty ((0,T);W^{2,\infty }(\Omega ))} \Big ) \le C_{\tau ,T,p}, \end{gathered} \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<\infty $$\end{document} . As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} , we have the following limits

$$[eqn]$$

and the limit vector (n, c, w) is the unique global classical solution to the indirect signalling parabolic-elliptic system (1.7)-(1.8).

As a by-product of the proof of Theorem 1.1, we have the following convergences, which also explain the mechanism of the PES

$$[eqn]$$

Up to now, we have only obtained the weak convergence for the equation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} due to a lack of uniform regularity information of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t w_\varepsilon $$\end{document} . We show that this strong convergence will be a consequence of the next part, where the accuracy of the PES provided in Theorem 1.1 is investigated. By subtracting the corresponding equations of solution components of the systems (1.1)-(1.3) and (1.7)-(1.8), we see that the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\widetilde{n}_{\varepsilon },\widetilde{c}_{\varepsilon },\widetilde{w}_{\varepsilon }):= (n_{\varepsilon }-n,c_{\varepsilon }-c, w_{\varepsilon }-w)$$\end{document} is the solution of the so-called rate system

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t \widetilde{n}_{\varepsilon }& \hspace{-0.2cm}=\hspace{-0.2cm}& \Delta \widetilde{n}_{\varepsilon }- \nabla \cdot (\widetilde{n}_{\varepsilon }\nabla c_\varepsilon + n\nabla \widetilde{c}_{\varepsilon }) & \text {in } \Omega _\infty , \\ \varepsilon \partial _t \widetilde{c}_{\varepsilon }& \hspace{-0.2cm}=\hspace{-0.2cm}& \Delta \widetilde{c}_{\varepsilon }- \widetilde{c}_{\varepsilon }+ \widetilde{w}_{\varepsilon }- \varepsilon \partial _t c & \text {in } \Omega _\infty , \\ \varepsilon \partial _t \widetilde{w}_{\varepsilon }& \hspace{-0.2cm}=\hspace{-0.2cm}& \tau \Delta \widetilde{w}_{\varepsilon }- \widetilde{w}_{\varepsilon }+ \widetilde{n}_{\varepsilon }- \varepsilon \partial _t w & \text {in } \Omega _\infty , \end{array} \right. \end{aligned}$$\end{document}

which is subjected to the boundary conditions

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \widetilde{n}_{\varepsilon }}{\partial \nu } = \frac{\partial \widetilde{c}_{\varepsilon }}{\partial \nu } = \frac{\partial \widetilde{w}_{\varepsilon }}{\partial \nu } = 0 \quad \text {on } \Gamma _\infty , \end{aligned}$$\end{document}

and the initial value condition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\widetilde{n}_{\varepsilon }(0), \widetilde{c}_{\varepsilon }(0), \widetilde{w}_{\varepsilon }(0)) = (0, c_0 - c(0), w_0 - w(0)). \end{aligned}$$\end{document}

It is obvious to see that c(0) and w(0) are not given a priori, and they may be well different from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_0$$\end{document} , respectively. These missing initial values can only be recovered, thanks to the last two equations in (1.7)-(1.8), as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w(x,0) = (-\tau \Delta + I)^{-1}n_0, \quad c(x,0) = (- \Delta + I)^{-1}w(x,0). \end{aligned}$$\end{document}

This difference in the initial values is referred to as the initial layer. It has been usually assumed to be zero in the literature, see e.g. [26]. However, this turns out to be important in studying the accuracy of the PES (or IDS), which is evidenced in the recent work [40], where the effect of the initial layer has been carefully analysed for the PES of a competitive prey-predator chemotaxis system. This effect is especially relevant when the original initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_0, c_0, w_0)$$\end{document} do not lie on the critical manifold, which is defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal C_{\textsf{PES}}:= \Big \{(n,c,w) \in L^2(\Omega ) \times H^2(\Omega )^2: \, (\Delta c - c + w , \, \tau \Delta w - w + n) =(0,0) \Big \}. \end{aligned}$$\end{document}

We define the distance from the initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_0,c_0,w_0)$$\end{document} to the critical manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_{\textsf{PES}} $$\end{document} with respect to the topology \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{k,p}(\Omega ) \times W^{l,p}(\Omega )$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{dist}^{k,l}_p[(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] := {\sqrt{\Vert - \Delta c_0 + c_0 -w_0\Vert _{W^{k,p}(\Omega )}^2 + \Vert - \tau \Delta w_0 + w_0-n_0\Vert _{W^{l,p}(\Omega )}^2}}\,, \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k,l\in \mathbb N$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p\le \infty $$\end{document} . When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=l=0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2$$\end{document} , we conveniently write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}:= \textrm{dist}^{0,0}_2$$\end{document} . By using the following representations of the inverse operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta + I)^{-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\tau \Delta + I)^{-1}$$\end{document} , see e.g. [40],

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} \displaystyle \widetilde{c}_{\varepsilon }(x,0) \; = \int _0^{\infty } e^{s(\Delta - I)}[-\Delta c_0(x) + c_0(x) -w_0(x)] ds , & x \in \Omega , \\ \displaystyle \widetilde{w}_{\varepsilon }(x,0) = \int _0^{\infty } e^{s(\tau \Delta - I)}[-\tau \Delta w_0(x) + w_0(x) - n_0(x)] ds , & x \in \Omega , \end{array}\right. \end{aligned}$$\end{document}

we can estimate these the initial layers by the distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}^{1,2}_p [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]$$\end{document} , see Lemma 3.1. Then, we can employ the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} estimates in Theorem 1.1 to obtain for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le k\in \mathbb N$$\end{document} (see Lemma 3.3),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{d}{dt} \int _{\Omega }\widetilde{n}_{\varepsilon }^{2k}(t) \le -\dfrac{2k-1}{k} \int _{\Omega }|\nabla \widetilde{n}_{\varepsilon }^{k}|^2 + C_{k, T}\int _{\Omega }\widetilde{n}_{\varepsilon }^{2k} + C_{k, T}\int _{\Omega }|\nabla \widetilde{c}_{\varepsilon }|^2, \end{aligned}$$\end{document}

to test the equations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{c}_{\varepsilon },\widetilde{w}_{\varepsilon }$$\end{document} , and apply the fundamental differential inequality given in Lemma A.6 to obtain convergence rates as follows.

Theorem 1.2

( Convergence rates and the initial layer’s effect) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 4$$\end{document} , and fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} . For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\varepsilon ,c_\varepsilon ,w_\varepsilon )$$\end{document} be the global classical solution to the system (1.1)-(1.3), given by Theorem 2.1.

a) Assuming that the distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}^{2,1}_2 [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]$$\end{document} is finite. Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{n}_{\varepsilon }\Vert _{L^{\infty }((0,T);L^2(\Omega ))} + \Vert \widetilde{n}_{\varepsilon }\Vert _{L^{2}((0,T); H^1(\Omega ))}&\le C_T \big ( \varepsilon + \sqrt{\varepsilon } \, \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big ), \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{w}_{\varepsilon }\Vert _{L^{\infty }((0,T);H^1(\Omega ))} + \Vert \widetilde{w}_{\varepsilon }\Vert _{L^{2}((0,T);H^2(\Omega ))} \le \,&\, C_{T,\tau } \big ( \varepsilon + \textrm{dist}^{0,1}_2 [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big ), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{c}_{\varepsilon }\Vert _{L^{\infty }((0,T);H^2(\Omega ))} + \Vert \widetilde{c}_{\varepsilon }\Vert _{L^{2}((0,T);H^3(\Omega ))} \; \le \,&\, C_{T,\tau } \big ( \varepsilon + \textrm{dist}^{2,1}_2 [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big ) . \end{aligned}$$\end{document}

b) Assuming that the distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}^{4,2}_p [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]$$\end{document} is finite for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le p<\infty $$\end{document} . Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{n}_{\varepsilon }\Vert _{L^{\infty }((0,T); L^{p}(\Omega ))}&\le C_{p,T,\tau } \left( \varepsilon ^{\frac{2}{p}} + \varepsilon ^{\frac{1}{p}} \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]\big )^{\frac{2}{p}} \right) , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{lllll} \displaystyle \Vert \widetilde{w}_{\varepsilon }\Vert _{L^p((0,T);W^{2,p}(\Omega ))} & \hspace{-0.2cm} \displaystyle \le C_{p,\tau ,T} \left( \varepsilon ^{\frac{2}{p}} + \varepsilon ^{\frac{1}{p}} \big ( \textrm{dist}^{0,2}_p [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]\big )^{\frac{2}{p}} \right) , \\ \displaystyle \Vert \widetilde{c}_{\varepsilon }\Vert _{L^p((0,T);W^{4,p}(\Omega ))} & \hspace{-0.2cm} \displaystyle \le C_{p,\tau ,T} \left( \varepsilon ^{\frac{2}{p}} + \varepsilon ^{\frac{1}{p}} \big ( \textrm{dist}^{4,2}_p [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]\big )^{\frac{2}{p}} \right) . \end{array} \end{aligned}$$\end{document}

Remark 1.1

• In the above estimates, the general constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T,\tau }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{p,T,\tau }$$\end{document} may tend to infinity as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \rightarrow 0$$\end{document} .
• Thanks to the estimate (1.22), the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \widetilde{n}_{\varepsilon }\Vert _{L^{\infty }((0,T);L^2(\Omega ))}$$\end{document} is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon )$$\end{document} if the distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]$$\end{document} is at least of the order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{\varepsilon }$$\end{document} . Even if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]$$\end{document} is large (i.e., the system starts far away from the critical manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_{\textsf{PES}}$$\end{document} ), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} always converges to n in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }((0,T);L^2(\Omega ))$$\end{document} at least in the order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\sqrt{\varepsilon })$$\end{document} . However, this is not true for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} . In [40], it has been shown numerically that, if a system starts far away from its critical manifold, then the slow evolution’s components do not converge to their expected limits in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^2(\Omega ))$$\end{document} , and, in contrast, the distances between the solutions can be even sufficiently large.
• Since the initial conditions are a major difference between the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} -dependent and limiting systems, a non-zero distance from the initial data to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_{\textsf{PES}}$$\end{document} corresponds to an initial layer. Therefore, Theorem 1.2 also claims that the parabolic-elliptic system (1.7)-(1.8) is a “good" approximation of the parabolic-parabolic system (1.1)-(1.3) whenever there is no initial layer, which is recently discussed in [26]. This suggests that skipping the slow time evolution should be associated with well-prepared initial data.

As discussed after Theorem (), we see that the weak convergence in (1.14) can, in fact, be proved in the strong topology. The following corollary is understood as the strong convergence to the critical manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\textsf {PES}}$$\end{document} .

Corollary 1

(Strong convergence to the critical manifold) For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\varepsilon ,c_\varepsilon ,w_\varepsilon )$$\end{document} be the global classical solution to the system (1.1)-(1.3). Then it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta c_\varepsilon - c_\varepsilon + w_\varepsilon \Vert _{L^2((0,T);H^1(\Omega ))} + \Vert \tau \Delta w_\varepsilon - w_\varepsilon + n_\varepsilon \Vert _{L^2(\Omega _T)} \le C_{T,\tau } \sqrt{\varepsilon } . \end{aligned}$$\end{document}

Furthermore, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}^{0,1}_2 [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]= O(\varepsilon )$$\end{document} then we have the improved convergence rate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\Vert \Delta c_\varepsilon - c_\varepsilon + w_\varepsilon \Vert _{L^2((0,T);H^1(\Omega ))} + \Vert \tau \Delta w_\varepsilon - w_\varepsilon + n_\varepsilon \Vert _{L^2(\Omega _T)} \le C_{T,\tau } {\varepsilon }.} \end{aligned}$$\end{document}

Proof

By the triangle inequality and the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \Delta w - w + n=0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \tau \Delta w_\varepsilon - w_\varepsilon + n_\varepsilon \Vert _{L^2(\Omega _T)} \le \,&\, {\Vert \tau \Delta \widetilde{w}_{\varepsilon }- \widetilde{w}_{\varepsilon }+ \widetilde{n}_{\varepsilon }\Vert _{L^2(\Omega _T)}}\\ \le \,&\, {\tau \Vert \widetilde{w}_{\varepsilon }\Vert _{L^2(0,T;H^2(\Omega ))} + \Vert \widetilde{w}_{\varepsilon }\Vert _{L^2(\Omega _T)} + \Vert \widetilde{n}_{\varepsilon }\Vert _{L^2(\Omega _T)}} \\ \le \,&\, C_{T,\tau } \big ( \varepsilon + \sqrt{\varepsilon } \textrm{dist}^{0,1}_2 [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big ) \end{aligned}$$\end{document}

thanks to (1.25) and (1.26). The convergence for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon $$\end{document} follows similarly. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Our second main results concerning rigorous IDS for (1.1)-(1.3) will be presented in Theorem 1.3. More precisely, we study the limit as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa =( \varepsilon ,\tau ) \rightarrow (0,0)$$\end{document} , or in other words, both parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} tend to zero at the same time. Here, the subscript \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\kappa ,c_\kappa ,w_\kappa )$$\end{document} is used to indicate the dependence of the solution on both parameters. We formally expect

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & (n_\kappa ,c_\kappa ,w_\kappa ) \rightarrow (n,c,w) \; \text {and} \; (\varepsilon \partial _t c_\kappa , \varepsilon \partial _t w_\kappa - \tau \Delta w_\kappa ) \nonumber \\ & \quad =(\Delta c_\kappa - c_\kappa + w_\kappa , - w_\kappa + n_\kappa ) \rightarrow (0,0), \end{aligned}$$\end{document}

and subsequently, at the limit level \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=n$$\end{document} . Therefore, the vector (n, c) is expected to be the solution to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t n = \Delta n - \nabla \cdot (n \nabla c) & \text {in } \Omega \times (0,\infty ) , \\ \Delta c - c + n = 0 & \text {in } \Omega \times (0,\infty ), \\ \dfrac{\partial n}{\partial \nu } = \dfrac{\partial c}{\partial \nu } = 0 & \text {on } \Gamma \times (0,\infty ), \\ n|_{t=0}=n_{0} & \text {on } \Omega , \end{array} \right. \end{aligned}$$\end{document}

which describes a direct signalling mechanism and is the well-known Keller-Segel system. Particularly, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =\varepsilon $$\end{document} , or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ,\varepsilon $$\end{document} are given in the same time scale, the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} can be rewritten as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t w_\varepsilon - \Delta w_\varepsilon = - \frac{1}{\varepsilon } \left( w_\varepsilon - n_\varepsilon \right) \end{aligned}$$\end{document}

in which the kinetics of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} is on a much faster time scale compared to its evolution and diffusion. The limit as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} then falls into the topic of fast reaction limits, which has usually been studied in reaction-diffusion systems with fast interaction, see e.g. [6, 30, 34, 42], and recently in chemotaxis systems [23, 25]. To rigorously prove IDS, similarly to Theorem 1.1, it is important to control the Lyapunov functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal E(n_\kappa ,c_\kappa )$$\end{document} as well as to obtain the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} estimates in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} , and therefore, we face similar challenges as in the first part. Furthermore, due to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \rightarrow 0^+$$\end{document} , the Lyapunov structure from Theorem 1.1 only gives the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} boundedness in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);H^1(\Omega ))$$\end{document} since the term of second order derivatives of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\kappa $$\end{document} now depends explicitly on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} . Obtaining uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} estimates is quite tricky since now both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} can be degenerate. Our idea is to adapt the bootstrap argument proposed in [30]. The starting point in this argument is given in Lemma 4.3, where we show there is a small constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \text {ess sup}_{t\in (0,T)} \int _{\Omega }n_\kappa ^{1+\frac{\delta }{2}} (t) + \Vert n_\kappa \Vert _{L^{2+\delta }(\Omega _T)} + \iint _{\Omega _T} n_\kappa ^{\frac{\delta }{2}-1} |\nabla n_\kappa |^2 \right) \le C_{T}. \end{aligned}$$\end{document}

Then, based on a combination of the heat regularisation, the Gagliardo-Nirenberg inequality, as well as the maximal regularity with slow evolution, we obtain a recursive increasing sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p_j\}_{j=0,1,...}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0:=1+\delta /2$$\end{document} satisfying: if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \Vert n_\kappa \Vert _{L^{2p_j}(\Omega _T)} \right) \le C_T, \end{aligned}$$\end{document}

then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) + \Vert n_\kappa \Vert _{L^{2p_{j+1}}(\Omega _T)} \right) \le C_{T,p_{j+1}}, \end{aligned}$$\end{document}

see Lemma 4.4. This is sufficient to perform a bootstrap argument to have the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(\Omega _T)$$\end{document} -boundedness for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p < \infty $$\end{document} that turns into the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} -boundedness due to the use of the Neumann heat semigroup. Finally, the convergence rate is obtained similarly to Theorem 1.2 by tracking carefully the dependence of all constants on both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , as well as the distance from the initial data to the critical manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_{\textsf{IDS}}$$\end{document} , which is defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal C_{\textsf{IDS}}:= \Big \{(n,c,w) \in L^2(\Omega ) \times H^2(\Omega ) \times L^2(\Omega ) : \, (\Delta c - c + w , - w + n) =(0,0) \Big \}. \end{aligned}$$\end{document}

The distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}^{k,l}_p [(n_0,c_0,w_0);\mathcal C_{\textsf{IDS}}]$$\end{document} is defined similarly to (1.20) due to the replacement of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_{\textsf{PES}}$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal C_{\textsf{IDS}}$$\end{document} .

Theorem 1.3

(IDS for (1.1)) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=1,2$$\end{document} . Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_0,c_0,w_0)$$\end{document} is complied with Assumption 1.1, and furthermore in the critical dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} M {:= \int _{\Omega }n_0} < 4 \pi . \end{aligned}$$\end{document}

For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa = (\varepsilon ,\tau )\in (0,\infty )^2$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\kappa ,c_\kappa ,w_\kappa )$$\end{document} be the global classical solution to the system (1.1)-(1.3), given by Theorem 2.1. Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<T<\infty $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \sup _{\kappa \in (0,\infty )^2} \Big ( \Vert n_\kappa \Vert _{C^{\gamma ,\gamma /2} (\overline{\Omega }\times [0,T])} + \Vert n_\kappa \Vert _{L^2((0,T);H^1(\Omega ))} \Big ) \le C_{T},\\ \sup _{\kappa \in (0,\infty )^2} \Big ( \Vert w_\kappa \Vert _{L^\infty (\Omega _T)} + \Vert w_\kappa \Vert _{L^2((0,T);H^{1}(\Omega ))} \Big ) \le C_{T},\\ \sup _{\kappa \in (0,\infty )^2} \Big ( \Vert c_\kappa \Vert _{L^\infty ((0,T);W^{1,\infty }(\Omega ))} + \Vert c_\kappa \Vert _{L^p((0,T);W^{2,p}(\Omega ))} \Big ) \le C_{T,p}. \end{aligned} \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<\infty $$\end{document} . As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa = (\varepsilon ,\tau )\rightarrow (0,0)$$\end{document} , we have the following limits

$$[eqn]$$

and the limit vector (n, c) is the unique global classical solution to the direct signalling parabolic-elliptic system (1.29). Moreover, assuming that the distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}^{1,0}_2 [(n_0,c_0,w_0);\mathcal C_{\textsf{IDS}}]$$\end{document} is finite. Then, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\kappa | = \varepsilon + \tau $$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{n}_{\varepsilon }\Vert _{L^{\infty }((0,T);L^2(\Omega ))} + \Vert \widetilde{n}_{\varepsilon }\Vert _{L^{2}((0,T); H^1(\Omega ))}&\le C_T \left( |\kappa | + \sqrt{|\kappa |} \, \textrm{dist}[(n_0,c_0,w_0);\mathcal C_{\textsf{IDS}}] \right) , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{w}_{\varepsilon }\Vert _{L^{\infty }((0,T);L^2(\Omega ))} + \Vert \widetilde{w}_{\varepsilon }\Vert _{L^{2}((0,T);H^1(\Omega ))}&\le C_{T} \Big ( |\kappa | + \textrm{dist}[(n_0,c_0,w_0);\mathcal C_{\textsf{IDS}}] \Big ), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{c}_{\varepsilon }\Vert _{L^{\infty }((0,T);H^1(\Omega ))} + \Vert \widetilde{c}_{\varepsilon }\Vert _{L^{2}((0,T);H^2(\Omega ))}&\le C_{T} \Big ( |\kappa | + \textrm{dist}^{1,0}_2[(n_0,c_0,w_0);\mathcal C_{\textsf{IDS}}] \Big ). \end{aligned}$$\end{document}

The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =0$$\end{document} was investigated in [25], where only the convergence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} to n as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} had been showed in a strong sense while $[eqn]$ weakly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^4((0,T);W^{1,4}(\Omega ))$$\end{document} and weakly-star in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);H^2(\Omega ))$$\end{document} and $[eqn]$ weakly-star in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} . Our results improve those of [25] by proving this convergence in the strong topology, and furthermore provide the convergence rate. Similarly to Corollary 1, we have the following strong convergence to the critical manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{\textsf{IDS}}$$\end{document} .

Corollary 2

(Strong convergence to the critical manifold) For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa = (\varepsilon ,\tau )\in (0,\infty )^2$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\kappa ,c_\kappa ,w_\kappa )$$\end{document} be the globally classical solution to the system (1.1)-(1.3). Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta c_\kappa - c_\kappa + w_\kappa \Vert _{L^2((0,T);H^1(\Omega ))} + \Vert - w_\kappa + n_\kappa \Vert _{L^2(\Omega _T)} \le C \sqrt{|\kappa |}. \end{aligned}$$\end{document}

The rest of this paper is organised as follows: In Section 2, we rigorously simplify from (1.1)-(1.3) to (1.7)-(1.8) in which both subcritical case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 3$$\end{document} and critical case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} are considered. The accuracy of this simplification is studied in Section 3. In Section 4, the analysis of the indirect-direct simplification from (1.1)-(1.3) to (1.29), as well as its accuracy, will be investigated. Finally, we place some auxiliary results in the Appendix A.

Rigorous parabolic-elliptic simplification

We start this section by the global existence and boundedness of solutions to (1.1)-(1.3) for fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} , which is done in [11]. We remark that the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\varepsilon ,\tau }$$\end{document} in the following theorem may tend to infinity as either \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \rightarrow 0$$\end{document} .

Theorem 2.1

([11, Theorem 1.1]) Suppose that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n_0, \, c_0, \,w_0 \ge 0 \text { on } \overline{\Omega }, \text { and } n_0 \in C(\bar{\Omega }), \, c_0, w_0 \in C^2(\overline{\Omega }). \end{aligned}$$\end{document}

For each pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varepsilon ,\tau ) \in (0,\infty )^2$$\end{document} , System (1.1)-(1.3) admits a unique classical positive solution (n, c, w) which exists globally in time. Moreover, it satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{t \in [0,\infty )} \Big {(} \Vert n(t)\Vert _{L^{\infty }(\Omega )} + \Vert c(t)\Vert _{W^{2,\infty }(\Omega )} + \Vert w(t)\Vert _{W^{2,\infty }(\Omega )} \Big {)} \le C_{\varepsilon ,\tau } < \infty . \end{aligned}$$\end{document}

Multiple time scale Lyapunov functional

By integrating the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} and using the homogeneous Neumann boundary condition, we have the conservation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega n_\varepsilon (x,t) = \int _\Omega n_0(x) = M, \quad \text {for all } t\ge 0, \end{aligned}$$\end{document}

which also reads that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^1(\Omega ))$$\end{document} . However, this regularity is not sufficient to gain necessary estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon ,c_\varepsilon $$\end{document} and then improve again the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} regularity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} . In this part, we present an a priori estimate for solutions by considering a Lyapunov functional according to the system structure. Since the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} can be rewritten as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial _t n_\varepsilon = \nabla \cdot \big ( n_\varepsilon \nabla (\log n_\varepsilon -c_\varepsilon ) \big ), $$\end{document}

we multiply two sides by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\log n_\varepsilon - c_\varepsilon )$$\end{document} and integrate over the spatial domain to get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega \partial _t n_\varepsilon (\log n_\varepsilon -c_\varepsilon ) = - \int _\Omega n_\varepsilon |\nabla (\log n_\varepsilon -c_\varepsilon )|^2 . \end{aligned}$$\end{document}

This suggests considering the Lyapunov functional below for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} E(n_\varepsilon )=\int _\Omega n_\varepsilon (\log n_\varepsilon - c_\varepsilon ), \end{aligned}$$\end{document}

which, after differentiating in time and taking into account that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _\Omega \partial _t n_\varepsilon =0$$\end{document} , gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} E(n_\varepsilon ) = - \int _\Omega n_\varepsilon |\nabla (\log n_\varepsilon -c)|^2 - \int _\Omega n_\varepsilon \partial _t c_\varepsilon . \end{aligned}$$\end{document}

An estimate for this type of functional was established corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =0$$\end{document} in [25, Section 4.1]. The analysis in our case is significantly more challenging since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 4$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} is the critical dimension. Concerning the last term of (2.4), we have the following computations.

Lemma 2.1

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} , it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} - \int _\Omega n_\varepsilon \partial _t c_\varepsilon = \,&\, - \frac{d}{dt} \int _\Omega \left( \frac{1}{2}|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 + \frac{\tau }{2} |\Delta c_\varepsilon |^2 + \frac{1+\tau }{2} |\nabla c_\varepsilon |^2 + \frac{1}{2} c_\varepsilon ^2 \right) \\ \,&\, - \frac{1+\tau }{\varepsilon } \int _{\Omega }|\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 - \frac{2}{\varepsilon } \int _{\Omega }|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 . \end{aligned} \end{aligned}$$\end{document}

Proof

Using the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{rlll} \varepsilon \partial _t w_\varepsilon & =& \varepsilon ^2 \partial _{tt}^2 c_\varepsilon - \varepsilon \Delta \partial _t c_\varepsilon + \varepsilon \partial _t c_\varepsilon , \\ \tau \Delta w_\varepsilon & =& \tau \varepsilon \Delta \partial _t c_\varepsilon - \tau \Delta ^2 c_\varepsilon + \tau \Delta c_\varepsilon , \\ w_\varepsilon & =& \varepsilon \partial _t c_\varepsilon - \Delta c_\varepsilon + c_\varepsilon . \end{array}\right. \end{aligned}$$\end{document}

Then, we imply from the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n_\varepsilon = \varepsilon ^2 \partial _{tt}^2 c_\varepsilon - (1+\tau )\varepsilon \Delta \partial _t c_\varepsilon + 2\varepsilon \partial _t c_\varepsilon + \tau \Delta ^2 c_\varepsilon - (1+\tau )\Delta c_\varepsilon + c_\varepsilon . \end{aligned}$$\end{document}

Therefore, due to the integration by parts,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \int _\Omega n_\varepsilon \partial _t c_\varepsilon&= - \int _\Omega \Big ( \varepsilon ^2 \partial _{tt}^2 c_\varepsilon - (1+\tau )\varepsilon \Delta \partial _t c_\varepsilon + 2\varepsilon \partial _t c_\varepsilon + \tau \Delta ^2 c_\varepsilon - (1+\tau )\Delta c_\varepsilon + c_\varepsilon \Big ) \partial _t c_\varepsilon \\&= - \frac{d}{dt} \left( \frac{\varepsilon ^2}{2} \int _\Omega |\partial _tc_\varepsilon |^2 + \frac{\tau }{2} \int _\Omega |\Delta c_\varepsilon |^2 + \frac{1+\tau }{2} \int _\Omega |\nabla c_\varepsilon |^2 + \frac{1}{2} \int _\Omega c_\varepsilon ^2 \right) \\&\hspace{0.45cm} - \left( (1+\tau )\varepsilon \int _\Omega |\nabla \partial _t c_\varepsilon |^2 + 2\varepsilon \int _\Omega |\partial _t c_\varepsilon |^2 \right) . \end{aligned}$$\end{document}

By using the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} at the last step, we obtain (2.5). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The time derivatives appearing above suggest that a combination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon (\log n_\varepsilon - c_\varepsilon )$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 + \frac{\tau }{2} |\Delta c_\varepsilon |^2 + \frac{1+\tau }{2} |\nabla c_\varepsilon |^2 + \frac{1}{2} c_\varepsilon ^2$$\end{document}

forms the relevant structure of a multiple time scale Lyapunov functional for the whole system. The following lemma is a direct consequence of Lemma 2.1 and the identity (2.4).

Lemma 2.2

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} , it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \mathcal {E}(n_{\varepsilon }(t),c_{\varepsilon }(t)) = - \mathcal {D}(n_{\varepsilon }(t), c_{\varepsilon }(t)) \le 0 \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}(n_{\varepsilon },c_{\varepsilon })$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}(n_{\varepsilon }, c_{\varepsilon })$$\end{document} are defined in (1.9) and (1.10), respectively.

Lemma 2.2 suggests an estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);H^2(\Omega ))$$\end{document} uniformly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , as well as in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);H^1(\Omega ))$$\end{document} uniformly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} . However, we note here that the lower boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}$$\end{document} has not been guaranteed since it contains \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-n_\varepsilon c_\varepsilon $$\end{document} . Therefore, to apply Lemma 2.2, a lower bound for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-n_\varepsilon c_\varepsilon $$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n_\varepsilon (\log n_\varepsilon - c_\varepsilon )$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1(\Omega _T)$$\end{document} must be established first. This will be done separately for the cases \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} in the following subsections, as the latter case is in the critical dimension and a different strategy needs to be employed.

The case of subcritical dimensions \documentclass[12pt]{minimal}

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			\begin{document}$$1\le N \le 3$$\end{document}1≤N≤3

Balancing the Lyapunov functional

Lemma 2.3

There exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C=C_{n_0,c_0,\Omega ,M}>0$$\end{document} independently of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ,\tau $$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon>0} \bigg ( \sup _{t>0} \int _\Omega \left( \frac{\tau }{4} |\Delta c_\varepsilon (t)|^2 + \frac{2+\tau }{4} |\nabla c_\varepsilon (t)|^2 + \frac{2-\tau }{4} c_\varepsilon ^2(t) \right) \bigg ) \le \frac{C}{\tau } , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \sup _{\varepsilon >0} \left( \frac{1}{\varepsilon } \iint _{\Omega _T} \Big ( |\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 + |\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 \Big ) \right) \le C_{\tau }. \end{aligned} \end{aligned}$$\end{document}

Proof

Under the assumption 1.1 on the initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_0,c_0)$$\end{document} , the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal E(n_0,c_0)$$\end{document} is clearly finite. By Lemma 2.2, for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {E}(n_{\varepsilon }(t),c_{\varepsilon }(t)) \le \mathcal {E}(n_0,c_0) - \int _0^t \mathcal D(n_\varepsilon (s),c_\varepsilon (s)) , \end{aligned}$$\end{document}

in more detail, which is equivalent to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \int _\Omega \left( (n_\varepsilon \log n_\varepsilon + e^{-1} ) + \frac{1}{2}|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 + \frac{\tau }{2} |\Delta c_\varepsilon |^2 + \frac{1+\tau }{2} |\nabla c_\varepsilon |^2 + \frac{1}{2} c_\varepsilon ^2 \right) \\ \le \mathcal E(n_0,c_0) - \int _0^t \mathcal D(n_\varepsilon (s),c_\varepsilon (s)) + e^{-1}|\Omega | + \int _\Omega n_\varepsilon c_\varepsilon . \end{aligned} \end{aligned}$$\end{document}

It is necessary to estimate the product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon c_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^1(\Omega ))$$\end{document} . By the Sobolev embedding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2(\Omega ) \hookrightarrow L^\infty (\Omega )$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega n_\varepsilon c_\varepsilon \le M \Vert c_\varepsilon \Vert _{L^{\infty }(\Omega )} \le C M \Vert c_\varepsilon \Vert _{H^{2}(\Omega )} \le \frac{\tau }{4} \Vert c_\varepsilon \Vert _{H^{2}(\Omega )}^2 + \frac{C^2M^2}{\tau }. \end{aligned}$$\end{document}

Therefore, we deduce from (2.9) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \int _\Omega \left( (n_\varepsilon \log n_\varepsilon + e^{-1} ) + \frac{1}{2}|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 + \frac{\tau }{4} |\Delta c_\varepsilon |^2 + \frac{2+\tau }{4} |\nabla c_\varepsilon |^2 + \frac{2-\tau }{4} c_\varepsilon ^2 \right) \\ + \int _0^t \mathcal D(n_\varepsilon (s),c_\varepsilon (s)) \le \mathcal E(n_0,c_0) + |\Omega | - M + \frac{C^2M^2}{\tau }, \end{aligned} \end{aligned}$$\end{document}

and hence, estimate (2.7) follows. In particular, by paying attention to the last two terms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal D(n_\varepsilon ,c_\varepsilon )$$\end{document} , we observe that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1+\tau }{\varepsilon } \iint _{\Omega _{t}}|\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 + \frac{2}{\varepsilon } \iint _{\Omega _{t}}|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 \le \frac{C}{\tau } \end{aligned}$$\end{document}

and obtain (2.8), where C depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_0,c_0, \Omega $$\end{document} and M and does not on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ,\tau $$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Uniform boundedness in sup-norms

Thanks to the Sobolev embedding, Lemma 2.3 implies that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^6(\Omega )^N$$\end{document} -norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla v$$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varepsilon ,t)$$\end{document} bounded. This will help us to obtain the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,\infty );L^2(\Omega )) \cap L^3(\Omega _T) \cap L^2((0,T);H^1(\Omega ))$$\end{document}

via testing its equation by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} , see Lemma 2.4. Moreover, in Lemma 2.5, this boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla c_\varepsilon $$\end{document} will show the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} via exploiting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p-L^q$$\end{document} estimates for the Neumann heat semigroup.

Lemma 2.4

It holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \left( \sup _{0<t<T} \int _{\Omega }n_\varepsilon ^2 + \iint _{\Omega _{T}}n_\varepsilon ^3 + \iint _{\Omega _{T}}|\nabla n_\varepsilon |^2 \right) \le C_\tau . \end{aligned}$$\end{document}

Proof

Multiplying the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} by itself, integrating by parts over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and using the Young inequality, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \int _{\Omega } n_{\varepsilon }^2 + \int _{\Omega } |\nabla n_{\varepsilon }|^2 \le \int _{\Omega } n_{\varepsilon }^2 |\nabla c_{\varepsilon }|^2 , \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} . Then, by the Hölder inequality,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \int _{\Omega } n_{\varepsilon }^2 + \int _{\Omega } |\nabla n_{\varepsilon }|^2 \le \Vert n_{\varepsilon }\Vert _{L^{3}(\Omega )}^2 \Vert \nabla c_{\varepsilon }\Vert _{L^6(\Omega )^N}^2. \end{aligned}$$\end{document}

Noting that the estimate (2.7) and the Sobolev embedding imply the uniform boundedness for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla c_{\varepsilon }$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);W^{1,6}(\Omega )^N)$$\end{document} , where the bound is proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\tau ^2$$\end{document} . Moreover, by applying the Gagliardo-Nirenberg interpolation inequality,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert n_{\varepsilon }\Vert _{L^3(\Omega )}^2 \Vert \nabla c_{\varepsilon }\Vert _{L^6(\Omega )^N}^2&\le \frac{C}{\tau ^2} \left( \frac{C}{\tau } \Vert \nabla n_{\varepsilon }\Vert ^{4/5}_{L^2(\Omega )^N} \Vert n_{\varepsilon }\Vert ^{1/5}_{L^1(\Omega )} + \Vert n_{\varepsilon }\Vert _{L^1(\Omega )} \right) ^2, \end{aligned}$$\end{document}

and by the Young inequality,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert n_{\varepsilon }\Vert _{L^3(\Omega )}^2 \Vert \nabla c_{\varepsilon }\Vert _{L^6(\Omega )^N}^2 \le \frac{C_M}{\tau ^4} \Vert \nabla n_{\varepsilon }\Vert ^{8/5}_{L^2(\Omega )^N} + \frac{C_M}{\tau ^2} \le \frac{1}{2} \int _{\Omega } |\nabla n_{\varepsilon }|^2 + \frac{C_M(\tau ^{18}+1)}{\tau ^{20}} . \end{aligned}$$\end{document}

Hence, estimate (2.11) is obtained directly from (2.12). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.5

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<T<\infty $$\end{document} , it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \Big ( \Vert n_\varepsilon \Vert _{L^\infty (\Omega _T)} \Big ) \le C_{\tau ,T}. \end{aligned}$$\end{document}

Proof

To prove the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} , we will estimate the quantity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Lambda _T := \sup _{0<t<T} \Vert n_{\varepsilon }(t)\Vert _{L^{\infty }(\Omega )}. \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3<p<6$$\end{document} and take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3}{2p}< \beta < \frac{1}{2}$$\end{document} . Then, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D((-\Delta +I)^\beta ) \hookrightarrow L^\infty (\Omega )$$\end{document} , thanks to Theorem 1.6.1 in [13]. Using the Duhamel formula and the estimate (A.1) for the heat Neumann semigroup, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert n_{\varepsilon }(t)\Vert _{L^{\infty }(\Omega )}&\le \Vert e^{t\Delta }n_0\Vert _{L^{\infty }(\Omega )} + \bigg \Vert \int _0^t e^{(t-s)\Delta } \nabla \cdot \left( n_{\varepsilon }(s) \nabla c_{\varepsilon }(s) \right) ds \bigg \Vert _{L^{\infty }(\Omega )} \\&\le \Vert e^{t\Delta }n_0\Vert _{L^{\infty }(\Omega )} + \int _0^t \big \Vert (-\Delta +I)^\beta e^{(t-s)\Delta } \nabla \cdot \left( n_{\varepsilon }(s) \nabla c_{\varepsilon }(s) \right) \big \Vert _{L^{p}(\Omega )} ds \\&\le \Vert n_0\Vert _{L^{\infty }(\Omega )} + C\int _0^t (t-s)^{-\beta -\frac{1}{2} - \eta } e^{-\lambda s} \Vert n_{\varepsilon }(s) \nabla c_{\varepsilon }(s)\Vert _{L^{p}(\Omega )^N} ds \end{aligned} \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta >0$$\end{document} , being chosen later. Using the Hölder inequality,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert n_{\varepsilon }(s) \nabla c_{\varepsilon }(s)\Vert _{L^p(\Omega )^N} \le C \Vert n_{\varepsilon }(s) \Vert _{L^{\frac{6p}{6-p}}(\Omega )} \Vert \nabla c_{\varepsilon }(s)\Vert _{L^{6}(\Omega )^N} \nonumber \\&\quad \le C \Vert n_{\varepsilon }(s) \Vert ^{\frac{7p-6}{6p}}_{L^{\infty }(\Omega )} \Vert n_{\varepsilon }(s) \Vert ^{\frac{6-p}{6p}}_{L^1(\Omega )} \Vert \nabla c_{\varepsilon }(s)\Vert _{L^6(\Omega )^N} \nonumber \le C_\tau \Lambda _T^{\frac{7p-6}{6p}}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sup _{t>0}\Vert \nabla c_{\varepsilon }(t)\Vert _{L^6(\Omega )^N}$$\end{document} is bounded due to estimate (2.7) and the Sobolev embedding for the dimensions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 3$$\end{document} . Combining the above estimates, we deduce that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Vert n_{\varepsilon }(t)\Vert _{L^{\infty }(\Omega )} \le C + C_\tau \Lambda _T^{\frac{7p-6}{6p}} \int _0^t (t-s)^{-\beta -\frac{1}{2} - \eta } e^{-\lambda s} ds. $$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta <1/2$$\end{document} , we can choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta <1/2-\beta $$\end{document} , which guarantees that the above improper integral is finite. Thus, we obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _T \le C + C_{\tau ,T} \Lambda _T^{(7p-6)/(6p)},$$\end{document} and therefore, the quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _T$$\end{document} must be bounded since its exponent on the right-hand side is strictly less than 1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.6

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} , it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \Big ( \Vert w_\varepsilon \Vert _{L^\infty ((0,T);W^{1,\infty }(\Omega ))} + \Vert \Delta w_\varepsilon \Vert _{L^p(\Omega _T)} \Big ) \le C_{\tau ,T} , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \Big ( \Vert c_\varepsilon \Vert _{L^\infty ((0,T);W^{2,\infty }(\Omega ))} \Big ) \le C_{\tau ,T} , \end{aligned}$$\end{document}

Proof

Thanks to the parabolic maximal regularity with slow evolution, cf. Lemma A.4, applied to the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta w_\varepsilon \Vert _{L^p(\Omega _T)} \le C_p \, \varepsilon ^{\frac{1}{p}} \Vert \Delta w_0\Vert _{L^p(\Omega )} + C_{p,\tau } \Vert n_\varepsilon \Vert _{L^p(\Omega _T)} \le C_{p,\tau ,T}, \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} . Now, using the Neumann heat semigroup, from the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} we can represent this component as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w_\varepsilon (t)= e^{\frac{1}{\varepsilon }t(\tau \Delta - I)} w_0 + \frac{1}{\varepsilon } \int _0^t e^{\frac{1}{\varepsilon }(t-s)(\tau \Delta - I)} n_\varepsilon (s) ds . \end{aligned}$$\end{document}

Therefore, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p_1 \le p_2 \le \infty $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0,1$$\end{document} , an application of estimate (A.2) shows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \nabla ^k w_\varepsilon (t)\Vert _{L^{p_2}(\Omega )} \le \left\| \nabla ^k e^{\frac{1}{\varepsilon }t(\tau \Delta - I)} w_0 \right\| _{L^{p_2}(\Omega )} + \frac{1}{\varepsilon } \left\| \int _0^t \nabla ^k e^{\frac{s}{\varepsilon }(\tau \Delta - I)} n_\varepsilon (t-s) ds \right\| _{L^{p_2}(\Omega )} \\&\quad \le C_\tau \Vert w_0 \Vert _{W^{k,p_2}(\Omega )} + \frac{C_\tau }{\varepsilon } \int _0^t e^{-\frac{s}{\varepsilon }} \min (s/\varepsilon ;1)^{-\frac{N}{2}\big (\frac{1}{p_1}-\frac{1}{p_2}\big )-\frac{k}{2}} \Vert n_\varepsilon (t-s)\Vert _{L^{p_1}(\Omega )} ds \\&\quad \le C_\tau \Vert w_0 \Vert _{W^{k,p_2}(\Omega )} + \frac{C_\tau }{\varepsilon } \Vert n_\varepsilon \Vert _{L^\infty ((0,T);L^{p_1}(\Omega ))} \int _0^t e^{-\frac{s}{\varepsilon }} \min (s/\varepsilon ;1)^{-\frac{N}{2}\big (\frac{1}{p_1}-\frac{1}{p_2}\big )-\frac{k}{2}} ds \end{aligned}$$\end{document}

using the uniform boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in Lemma 2.5. By taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1=p_2=\infty $$\end{document} , the latter term is bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T))$$\end{document} since it is obvious that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{\varepsilon } \int _0^t e^{-\frac{s}{\varepsilon }} \min (s/\varepsilon ;1)^{-\frac{k}{2}} ds \le \int _0^\infty e^{-s} \min (s;1)^{-\frac{k}{2}} ds \le C, \end{aligned}$$\end{document}

where the constant C does not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . This shows the uniform boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);W^{1,\infty }(\Omega ))$$\end{document} , which in combination with (2.15) shows (2.13).

For the component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} , it follows from its equation that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_\varepsilon (t)= e^{\frac{1}{\varepsilon }t(\Delta - I)} c_0 + \frac{1}{\varepsilon } \int _0^t e^{\frac{1}{\varepsilon }(t-s)(\Delta - I)} w_\varepsilon (s) ds , \end{aligned}$$\end{document}

and thus, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le q_1 \le q_2 \le \infty $$\end{document} , using estimate (A.2) again gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \Delta c_\varepsilon (t)\Vert _{L^{q_2}(\Omega )} \le C_\tau \Vert c_0 \Vert _{W^{2,q_2}(\Omega )} + \frac{C_\tau }{\varepsilon } \int _0^t e^{-\frac{s}{\varepsilon }} \min (s/\varepsilon ;1)^{-\frac{N}{2}\big (\frac{1}{q_1}-\frac{1}{q_2}\big )} \Vert \Delta w_\varepsilon (t-s)\Vert _{L^{q_1}(\Omega )} ds \\&\quad \le C_\tau \Vert c_0 \Vert _{W^{2,q_2}(\Omega )} + \frac{C_\tau }{\varepsilon } \Vert \Delta w_\varepsilon \Vert _{L^{q_1}(\Omega _T)} \left\| \int _0^t e^{-\frac{s}{\varepsilon }} \min (s/\varepsilon ;1)^{-\frac{N}{2}\big (\frac{1}{q_1}-\frac{1}{q_2}\big )} ds \right\| _{L^{q_1/(q_1-1)}((0,T))}. \end{aligned}$$\end{document}

Then, by choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_1 \gg 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_2 = \infty $$\end{document} , the latter temporal norm is finite, similarly to (2.16). Hence, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta c_\varepsilon $$\end{document} is uniformly bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} , and in the same way, we have the same conclusion for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} and its gradient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla c_\varepsilon $$\end{document} . Consequently, we obtain (2.14). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.7

There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \left( \Vert n_\varepsilon \Vert _{C^{\gamma ,\gamma /2}(\overline{\Omega }\times [0,T])} \right) \le C_{\tau ,T} . \end{aligned}$$\end{document}

Proof

Recalling for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\varepsilon ,c_\varepsilon ,w_\varepsilon )$$\end{document} is the globally classical solution to (1.1)-(1.3), so that it is continuous with respect to both time and space variables. Therefore, one can apply [35, Theorem 1.3 and Remark 1.4] or [20, Lemma 2.1, Part iv] to claim (2.17), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\tau ,T}$$\end{document} does not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} due to the uniform boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in Lemma 2.5 and of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} in Lemma 2.6. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Passage to the limit

Lemma 2.8

Assume that (n, c, w) is a globally weak solution to System (1.7)-(1.8) in the sense that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n \in C(\overline{\Omega }\times [0,T]) \cap L^\infty (\Omega _T) \cap L^2((0,T);H^1(\Omega )), \quad c,w\in L^2((0,T);H^1(\Omega )), \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} - \int _0^T \langle n , \partial _t \xi \rangle - \int _\Omega n_0 \xi (0) = \iint _{\Omega _{T}}(- \nabla n + n \nabla c) \cdot \nabla \xi , \\ \iint _{\Omega _{T}}( \nabla c \cdot \nabla \zeta + c\zeta ) = \iint _{\Omega _{T}}w\zeta , \\ \iint _{\Omega _{T}}( \tau \nabla w \cdot \nabla \zeta + w\zeta ) = \iint _{\Omega _{T}}n\zeta , \end{aligned} \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi , \zeta \in C_c^\infty (\overline{\Omega }\times [0,T))$$\end{document} . Then, it is the unique global classical solution to (1.7)-(1.8).

Proof

We first note that it is straightforward to check the initial condition in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega )$$\end{document} for n. Since the weak formulations for c and w are standard weak forms of the linear elliptic equations (specifically, the last two equations of (1.7)), it is obvious that they become the strong solutions to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \Delta c - c + w = 0 & \text {in } \Omega _\infty , \\ \tau \Delta w - w + n = 0 & \text {in } \Omega _\infty , \\ \partial _\nu c = \partial _\nu w = 0 & \text {on } \Gamma _\infty , \end{array} \right. \end{aligned}$$\end{document}

Using the representation of the inverse operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta + I)^{-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\tau \Delta + I)^{-1}$$\end{document} , for example, see [40, Appendix B], we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{llll} c(x,t) = \displaystyle \int _0^\infty e^{s(\Delta - I)} w(x,t) ds, & (x,t)\in \overline{\Omega }\times [0,T], \\ w(x,t) = \displaystyle \int _0^\infty e^{s(\tau \Delta - I)} n(x,t) ds, & (x,t)\in \overline{\Omega }\times [0,T]. \end{array} \right. \end{aligned}$$\end{document}

Therefore, the continuity of n implies the continuity of w and, then, of c. Consequently, the Hölder continuity of n is obtained using the results in [35, Theorem 1.3 and Remark 1.4] or in [20, Lemma 2.1, Part iv]. By the representation (2.21) again, we claim the Hölder continuity of w and c. This allows us to apply [20, Lemma 2.1, Part v] that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in C^{2,1}(\Omega \times (0,T))$$\end{document} , and so (n, c, w) becomes the unique classical solution to (1.7)-(1.8). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

In the following, we present the proof of Theorem 1.1 for setting subcritical dimensions.

Proof of Theorem 1.1 with subcritical dimensions N=1,2,3

We first note that boundedness (1.12) has been obtained in Lemmas 2.4, 2.6 and 2.7. In the following, we will prove the convergence of the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(n_\varepsilon ,c_\varepsilon ,w_\varepsilon )\}_{\varepsilon >0}$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} . Thanks to the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in the space of Hölder continuous functions obtained in Lemma 2.7, the Arzelà-Ascoli theorem yields that there exists a subsequence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{n_\varepsilon \}_{\varepsilon >0}$$\end{document} (being denoted by the same notation) such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n_\varepsilon \longrightarrow n \quad \text { strongly in } C(\overline{\Omega }\times [0,T]) \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} . Moreover, the estimate for this component in Lemma 2.4 also implies that

$$[eqn]$$

Testing the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in C_c^\infty (\overline{\Omega }\times [0,T))$$\end{document} , we derive

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \int _0^T \langle n_\varepsilon , \partial _t \xi \rangle - \int _\Omega n_0 \xi (0) = \iint _{\Omega _{T}}(- \nabla n_\varepsilon + n_\varepsilon \nabla c_\varepsilon ) \cdot \nabla \xi , \end{aligned}$$\end{document}

which, after using the convergence (2.22)-(2.23), shows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \int _0^T \langle n , \partial _t \xi \rangle - \int _\Omega n_0 \xi (0) = \iint _{\Omega _{T}}(- \nabla n + n \nabla c) \cdot \nabla \xi . \end{aligned}$$\end{document}

Next, we will consider the limits of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} . We note from the previous subsections that the uniform boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t c_\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t w_\varepsilon $$\end{document} is lacking. Therefore, the compactness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{c_\varepsilon \}_{\varepsilon >0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{w_\varepsilon \}_{\varepsilon >0}$$\end{document} does not make the Arzelà-Ascoli theorem or the Aubin-Lions lemma applicable. Thanks to Lemma 2.6,

$$[eqn]$$

Testing the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta \in C_c^\infty (\overline{\Omega }\times [0,T))$$\end{document} gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \varepsilon \int _\Omega w_\varepsilon (0)\zeta (0) - \varepsilon \iint _{\Omega _T} w_\varepsilon \partial _t \zeta + \iint _{\Omega _{T}}( \tau \nabla w_\varepsilon \cdot \nabla \zeta + w_\varepsilon \zeta ) = \iint _{\Omega _{T}}n_\varepsilon \zeta . \end{aligned}$$\end{document}

With the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} obtained in Lemma 2.6, we can pass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} to obtain the weak formulation for w in (2.19). Note that this can be done similarly for the component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} . Thus, the limit vector (n, c, w) is a globally weak solution to System (1.7)-(1.8) in the sense (2.18)-(2.19). Then, Lemma 2.8 yields that this solution becomes the unique globally classical solution of (1.7)-(1.8).

We now improve the convergence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon ,c_\varepsilon $$\end{document} to a strong sense, which will be basically based on the so-called energy equation method, see e.g. [3, 15], presented as follows. Recall that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _{T}}( \nabla w \cdot \nabla \zeta + w \zeta ) = \iint _{\Omega _{T}}n \zeta , \quad \text {for all } \zeta \in C_c^\infty (\overline{\Omega }\times [0,T)), \end{aligned}$$\end{document}

and for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} is sufficiently smooth since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n_\varepsilon ,c_\varepsilon ,w_\varepsilon )$$\end{document} is the globally classical solution to System (1.1)-(1.3). Due to an argument of dense spaces, we can choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} to be a test function in (2.25), which yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _{T}}( |\nabla w_\varepsilon |^2 + w_\varepsilon ^2) = \iint _{\Omega _{T}}n_\varepsilon w_\varepsilon - \frac{\varepsilon }{2} \int _{\Omega }(w_\varepsilon ^2 - w_0^2). \end{aligned}$$\end{document}

Then, choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi = w$$\end{document} in (2.26) gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{gathered} \iint _{\Omega _{T}}( |\nabla w|^2 + w^2) = \iint _{\Omega _{T}}n w, \end{gathered}\end{aligned}$$\end{document}

which is combined with (2.27) to show that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \Vert w_\varepsilon \Vert _{L^2((0,T);H^1(\Omega ))}^2 - \Vert w\Vert _{L^2((0,T);H^1(\Omega ))}^2 \right| \le \left| \iint _{\Omega _{T}}(n_\varepsilon w_\varepsilon - n w) \right| + \frac{\varepsilon }{2} \left| \int _{\Omega }(w_\varepsilon ^2 - w_0^2) \right| . \end{aligned}$$\end{document}

Using the convergence (2.22), (2.24), and the uniform boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T); L^2(\Omega ))$$\end{document} , cf. Lemma 2.6, the latter right-hand side tends to zero as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} . Therefore,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w_\varepsilon \Vert _{L^2((0,T);H^1(\Omega ))} \longrightarrow \Vert w\Vert _{L^2((0,T);H^1(\Omega ))}. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);H^1(\Omega ))$$\end{document} is uniformly convex, this implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w_\varepsilon \longrightarrow w \quad \text {strongly in } L^2((0,T);H^1(\Omega )). \end{aligned}$$\end{document}

Similarly, one can show the convergence

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_\varepsilon \longrightarrow c \quad \text {strongly in } L^2((0,T);H^1(\Omega )) , \end{aligned}$$\end{document}

and, for the same test functions as in (2.25),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{gathered} \iint _{\Omega _{T}}( \nabla c \cdot \nabla \xi + c \xi ) = \iint _{\Omega _{T}}w \xi . \end{gathered}\end{aligned}$$\end{document}

We obtain the convergence stated at (1.13) by collecting (2.22)-(2.23) and (2.28)-(2.29). While the first line in (1.14) is straightforward from estimate (2.8) in Lemma 2.3 by recalling \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon = \varepsilon \partial _t c_\varepsilon $$\end{document} , the second one is directly derived from (2.25) after integrating by parts in space. Since (n, c, w) is the unique solution to System (1.7)-(1.8), the above convergences hold for the whole sequences.

The case of critical dimension \documentclass[12pt]{minimal}

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			\begin{document}$$N=4$$\end{document}N=4

When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\le 3$$\end{document} , it is sufficient to use the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} -norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} and the embedding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2(\Omega )\hookrightarrow L^{\infty }(\Omega )$$\end{document} to control the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{\Omega }n_\varepsilon c_\varepsilon $$\end{document} . In the critical dimension, we ought to exploit the control of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{\Omega }n_\varepsilon \log n_\varepsilon $$\end{document} as well as an Adam-type inequality (see Lemmas A.2 and A.3) to balance the multiple time-scale entropy. This also leads to a restriction on the size of the initial mass M as (1.11).

Lemma 2.9

Assume that M satisfies (1.11). Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon>0} \left( \sup _{t>0} \int _{B_R} (n_\varepsilon \log n_\varepsilon + e^{-1}) + \sup _{t>0} {\Vert (\Delta -I)c_\varepsilon (t)\Vert _{L^2(B_R)}^2} \right) \le C_{\tau } , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \sup _{\varepsilon >0} \left( \frac{1}{\varepsilon } \iint _{B_R\times (0,\infty )} \Big ( |\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 + |\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 \Big ) \right) \le C_{\tau } . \end{aligned} \end{aligned}$$\end{document}

Proof

For any positive real numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =\eta (\alpha )>0$$\end{document} being chosen later, an application of the inequalities (A.5) and (A.7) gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{B_R} n_\varepsilon c_\varepsilon&\le \frac{1}{e} + \frac{1}{\alpha } \int _{B_R} n_\varepsilon \log n_\varepsilon + \frac{\Vert n_\varepsilon \Vert _{L^1(\Omega )}}{\alpha } \log \left( \int _{B_R} e^{\alpha c_\varepsilon } \right) \\&\le \frac{1}{e} + \frac{1}{\alpha } \int _{B_R} n_\varepsilon \log n_\varepsilon + \frac{M}{\alpha } \left[ \left( \frac{\alpha ^2}{128\pi ^2} + \eta \right) \Vert (\Delta -I)c_\varepsilon \Vert _{L^2(B_R)}^2 + C_{R,\eta ,\alpha } \right] . \end{aligned}$$\end{document}

Then, similarly to estimate (2.9), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\int _{B_R} \left( (n_\varepsilon \log n_\varepsilon + e^{-1}) + \frac{1}{2}|\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 + \frac{\tau }{2} |\Delta c_\varepsilon |^2 + \frac{1+\tau }{2} |\nabla c_\varepsilon |^2 + \frac{1}{2} c_\varepsilon ^2 \right) \\&\qquad + \int _0^t \hspace{-0.15cm} \int _{B_R} \Big ( n_\varepsilon |\nabla (\log n_\varepsilon - c_\varepsilon )|^2 + \frac{1+\tau }{\varepsilon } |\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 + \frac{2}{\varepsilon } |\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 \Big ) \\&\quad \le C + \frac{1}{e} + \frac{1}{\alpha } \int _{B_R} n_\varepsilon \log n_\varepsilon + \frac{M}{\alpha } \left[ \left( \frac{\alpha ^2}{128\pi ^2} + \eta \right) \Vert (\Delta -I) c_\varepsilon \Vert _{L^2(B_R)}^2 + C_{R,\eta ,\alpha } \right] , \end{aligned} \end{aligned}$$\end{document}

where C is the initial value of the entropy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal E$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M<64 \tau \pi ^2$$\end{document} , we can choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} and a sufficiently small number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta >0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{\alpha }< 1 \quad \text {and} \quad \frac{M}{\alpha } \left( \frac{\alpha ^2}{128\pi ^2} + \eta \right) < \frac{\tau }{2}, \end{aligned}$$\end{document}

which allows us to imply that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left( 1- \frac{1}{\alpha } \right) \int _{B_R} n_\varepsilon \log n_\varepsilon + \left( \frac{\tau }{2} - \frac{M}{\alpha } \left( \frac{\alpha ^2}{128\pi ^2} + \eta \right) \right) { \Vert (\Delta -I)c_\varepsilon \Vert _{L^2(B_R)}^2} \\&\quad + \int _0^t \hspace{-0.15cm} \int _{B_R} \Big ( \frac{1+\tau }{\varepsilon } |\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 + \frac{2}{\varepsilon } |\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon |^2 \Big ) \le C_{R,\eta ,\alpha ,M}, \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<t<\infty $$\end{document} . The estimates (2.30)-(2.31) are consequently obtained. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.10

Assume that M satisfies (1.11). Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \left( \iint _{B_R\times (0,T)} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } ds + \int _0^T \Vert n_\varepsilon \Vert _{L^{4/3}(B_R)}^2 ds \right) \le C_{T,\tau }. \end{aligned}$$\end{document}

Proof

This proof will be based on balancing a logarithmic energy below. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x >0$$\end{document} , let us denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x):= x \log x - x + 1$$\end{document} . By direct computations, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \int _{B_R} h(n_\varepsilon ) = - \int _{B_R} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } - \int _{B_R} n_\varepsilon \Delta c_\varepsilon , \end{aligned}$$\end{document}

which, after integrating over time, gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \int _{B_R} h(n_\varepsilon ) ds + \iint _{B_R\times (0,t)} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } ds \le \int _{B_R} h(n_0) - \underbrace{\iint _{B_R\times (0,t)} n_\varepsilon \Delta c_\varepsilon ds}_{=:\,-I_\varepsilon (t)} . \end{aligned} \end{aligned}$$\end{document}

In the remaining, we will control the quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\varepsilon (t)$$\end{document} using the norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T); L^{4/3}(\Omega ))$$\end{document} , and then balance the estimate (2.33) of the above logarithmic energy.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\textrm{Estimating}\,I_\varepsilon (t)}$$\end{document} : Using the equations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon ,w_\varepsilon $$\end{document} , we have the following computations

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \int _{B_R} n_\varepsilon \Delta c_\varepsilon&= \int _{B_R} n_\varepsilon (- \varepsilon \partial _t c_\varepsilon - c_\varepsilon + w_\varepsilon ) = - \varepsilon \int _{B_R} n_\varepsilon \partial _t c_\varepsilon - \int _{B_R} n_\varepsilon c_\varepsilon + \int _{B_R} n_\varepsilon w_\varepsilon \\&= - \varepsilon \int _{B_R} n_\varepsilon \partial _t c_\varepsilon - \int _{B_R} n_\varepsilon c_\varepsilon + \int _{B_R} (\varepsilon \partial _t w_\varepsilon - \tau \Delta w_\varepsilon + w_\varepsilon ) w_\varepsilon \\&= - \varepsilon \int _{B_R} n_\varepsilon \partial _t c_\varepsilon - \int _{B_R} n_\varepsilon c_\varepsilon + \int _{B_R} \left( \frac{\varepsilon }{2} \partial _t w_\varepsilon ^2 + \tau |\nabla w_\varepsilon |^2 + w_\varepsilon ^2 \right) \\&\le \varepsilon \Vert n_\varepsilon (t)\Vert _{L^{\frac{4}{3}}(B_R)} \Vert \partial _tc_\varepsilon \Vert _{L^{4}(B_R)} + \int _{B_R} \left( \frac{\varepsilon }{2} \partial _t w_\varepsilon ^2 + \tau |\nabla w_\varepsilon |^2 + w_\varepsilon ^2 \right) . \end{aligned}$$\end{document}

By the Young inequality, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varepsilon \Vert n_\varepsilon (t)\Vert _{L^{4/3}(B_R)} \Vert \partial _tc_\varepsilon \Vert _{L^{4}(B_R)} \le \frac{\varepsilon }{2} \Vert n_\varepsilon (t)\Vert _{L^{4/3}(B_R)}^2 + \frac{\varepsilon }{2} \Vert \partial _tc_\varepsilon \Vert _{L^{4}(B_R)}^2 , \end{aligned}$$\end{document}

and by the Sobolev embedding,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\varepsilon }{2} \Vert \partial _tc_\varepsilon \Vert _{L^{4}(B_R)}^2 \le C \varepsilon \left( \Vert \nabla \partial _tc_\varepsilon \Vert _{L^{2}(B_R)}^2 + \Vert \partial _tc_\varepsilon \Vert _{L^{2}(B_R)}^2 \right) . \end{aligned}$$\end{document}

Consequently, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} I_\varepsilon (t)&\le \frac{\varepsilon }{2} \int _0^t \Vert n_\varepsilon (s)\Vert _{L^{4/3}(B_R)}^2 ds + C\varepsilon \iint _{B_R\times (0,t)} \left( |\nabla \partial _sc_\varepsilon |^2 + |\partial _sc_\varepsilon |^2 \right) ds \\&\quad + \frac{\varepsilon }{2} \iint _{B_R\times (0,t)} \partial _s w_\varepsilon ^2 ds + \iint _{B_R\times (0,t)} \left( \tau |\nabla w_\varepsilon |^2 + w_\varepsilon ^2 \right) ds \\&\le \frac{\varepsilon }{2} \int _0^t \Vert n_\varepsilon (s)\Vert _{L^{4/3}(B_R)}^2 ds + C\varepsilon \iint _{B_R\times (0,t)} \left( |\nabla \partial _sc_\varepsilon |^2 + |\partial _sc_\varepsilon |^2 \right) ds \\&\quad + \frac{\varepsilon }{2} \int _{B_R} w_\varepsilon ^2 + \iint _{B_R\times (0,t)} \left( \tau |\nabla w_\varepsilon |^2 + w_\varepsilon ^2 \right) ds . \end{aligned} \end{aligned}$$\end{document}

Recalling the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial _tc_\varepsilon = (1/\varepsilon )(\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )$$\end{document} , the dissipation in Lemma 2.9 is rewritten as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varepsilon \iint _{B_R\times (0,t)} \Big ( |\nabla \partial _sc_\varepsilon |^2 ds + |\partial _sc_\varepsilon |^2 ds \Big ) \le C, \end{aligned}$$\end{document}

On the other hand, by applying Lemma A.5 to the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\varepsilon }{2} \int _{B_R} w_\varepsilon ^2 + \iint _{B_R\times (0,t)} \left( |\nabla w_\varepsilon |^2 + w_\varepsilon ^2 \right) ds \le \int _{B_R} w_0^2 + \frac{C}{\tau ^2} \int _0^t \Vert n_\varepsilon \Vert _{L^{\frac{4}{3}}(B_R)}^2 ds . \end{aligned}$$\end{document}

Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\varepsilon (t)$$\end{document} is estimated as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_\varepsilon (t)&\le \left( \frac{\varepsilon }{2} + \frac{C}{\tau ^2} \right) \int _0^t \Vert n_\varepsilon \Vert _{L^{\frac{4}{3}}(B_R)}^2 ds + C + \int _{B_R} w_0^2. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\text {Balancing the logarithmic energy}}$$\end{document} : We will apply Lemma A.1 to control the term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{0}^t\Vert n_\varepsilon \Vert _{L^{\frac{4}{3}}(B_R)}^2ds$$\end{document} . Due to the computation (2.33) and the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\varepsilon (t)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{B_R} h(n_\varepsilon ) ds + \iint _{B_R\times (0,t)} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } ds \le \left( \frac{3}{2} + \frac{C}{\tau ^2} \right) \int _0^t \Vert n_\varepsilon \Vert _{L^{\frac{4}{3}}(B_R)}^2 ds + C_{\tau }, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_\tau $$\end{document} includes the value of the logarithmic entropy at the initial time and the last two terms in the estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\varepsilon (t)$$\end{document} . By Lemma 2.9, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon \in L^\infty ((0,T);L\log L(B_R))$$\end{document} . Therefore, an application of Lemma A.1 gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert n_\varepsilon (t)\Vert _{L^{\frac{4}{3}}(B_R)}^2&\le \alpha \left( \int _{B_R} (n_\varepsilon (t) \log n_\varepsilon (t) + e^{-1}) \right) \int _{B_R} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } + C_\alpha \\&\le \alpha \left( \sup _{t>0} \int _{B_R} (n_\varepsilon (t) \log n_\varepsilon (t) + e^{-1}) \right) \int _{B_R} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } + C_\alpha \\&\le \frac{1}{2} \left( \frac{3}{2} + \frac{C}{\tau ^2} \right) ^{-1} \int _{B_R} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } + C_\tau , \end{aligned} \end{aligned}$$\end{document}

where we take a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \alpha \left( \sup _{t>0} \int _{B_R} (n_\varepsilon (t) \log n_\varepsilon (t) + e^{-1}) \right) \le \frac{1}{2} \left( \frac{3}{2} + \frac{C}{\tau ^2} \right) ^{-1}. \end{aligned}$$\end{document}

Hence, we can absorb the term including \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert n_\varepsilon \Vert _{L^{\frac{4}{3}}(B_R)}^2$$\end{document} in the estimate (2.34) into the left-hand side, which consequently implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{B_R} h(n_\varepsilon (t)) ds + \iint _{B_R\times (0,t)} \frac{|\nabla n_\varepsilon |^2}{n_\varepsilon } ds \le C_{\tau } t. \end{aligned}$$\end{document}

This directly shows the first estimate in (2.32). The second one follows immediately by integrating (2.35) with respect to t and using the first estimate. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.11

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} , it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \left( \sup _{0<t<T} \int _{B_R} n_\varepsilon ^p(t) + \iint _{B_R\times (0,T)} |\nabla n_\varepsilon |^2 \right) \le C_{p,T}. \end{aligned}$$\end{document}

Proof

Using the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} , one can check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \frac{d}{dt} \int _{B_R} n_\varepsilon ^p(t)&= -\frac{4(p-1)}{p} \int _{B_R} | \nabla n_\varepsilon ^{p/2}(t)|^2 + p(p-1) \int _{B_R} n_\varepsilon ^{p-1}(t) \nabla n_\varepsilon (t) \cdot \nabla c_\varepsilon (t) \\&\le - \frac{4(p-1)}{p} \int _{B_R} | \nabla n_\varepsilon ^{p/2}(t)|^2 + \frac{p(p-1)}{2} \int _{B_R} n_\varepsilon ^{p}(t) |\nabla c_\varepsilon (t)|^2 . \end{aligned} \end{aligned}$$\end{document}

To estimate this energy, we will control the latter term by the product of the integral of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon ^p$$\end{document} and a suitable norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla c_\varepsilon $$\end{document} . Indeed, using the Hölder, the Gagliardo-Nirenberg and the Young inequalities, it can be dealt with as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\int _{B_R} (n_\varepsilon ^{p/2}(t))^2 |\nabla c_\varepsilon (t)|^2 \le \Vert n_\varepsilon ^{p/2}(t)\Vert _{L^{\frac{8}{3}}(B_R)}^2 \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^2 \\&\quad \le \bigg (C \Vert n_\varepsilon ^{p/2}(t)\Vert _{L^{2}(B_R)}^{\frac{1}{2}} \Vert \nabla n_\varepsilon ^{p/2}(t)\Vert _{L^{2}(B_R)}^{\frac{1}{2}} + C \Vert n_\varepsilon ^{p/2}(t)\Vert _{L^{2}(B_R)} \bigg )^2 \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^2 \\&\quad \le \bigg (C \Vert n_\varepsilon ^{p/2}(t)\Vert _{L^{2}(B_R)} \Vert \nabla n_\varepsilon ^{p/2}(t)\Vert _{L^{2}(B_R)} + C \Vert n_\varepsilon ^{p/2}(t)\Vert _{L^{2}(B_R)}^2 \bigg ) \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^2 \\&\quad \le \frac{4}{p^2} \int _{B_R} | \nabla n_\varepsilon ^{p/2}(t)|^2 + C \left( \frac{p^2}{16} \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^4 + \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^2 \right) \int _{B_R} n^p_\varepsilon (t). \end{aligned} \end{aligned}$$\end{document}

Thus, we deduce from (2.37) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt} \int _{B_R} n_\varepsilon ^p(t)&\le - \frac{2(p-1)}{p} \int _{B_R} | \nabla n_\varepsilon ^{p/2}(t)|^2 + C_p \left( 1+ \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^4 \right) \int _{B_R} n^p_\varepsilon (t) . \end{aligned}$$\end{document}

It remains to estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla c_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^4((0,T);L^8(B_R))$$\end{document} , which will be done using Lemmas 2.9-2.10 and A.5. Indeed, thanks to the uniform boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);L^{4/3}(B_R))$$\end{document} obtained in Lemma 2.10, we can apply Lemma A.5 to have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{B_R\times (0,T)} |\nabla w_\varepsilon |^2 \le \int _{B_R} u_0^2 + \frac{C}{\tau ^2} \int _0^{T} \Vert n_\varepsilon \Vert _{L^{\frac{4}{3}}(B_R)}^2 \le C_{\tau ,T}. \end{aligned}$$\end{document}

On the other hand, by Lemma 2.9,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^T \hspace{-0.15cm} \int _{B_R} |\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 \le C_{\tau } \varepsilon . \end{aligned}$$\end{document}

Therefore, it follows from the uniform boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);H^2(B_R))$$\end{document} the triangle estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^T \hspace{-0.15cm} \int _{B_R} |\nabla \Delta c_\varepsilon |^2 \le C \int _0^T \hspace{-0.15cm} \int _{B_R} \Big ( |\nabla (\Delta c_\varepsilon - c_\varepsilon + w_\varepsilon )|^2 + |\nabla c_\varepsilon |^2 + |\nabla w_\varepsilon |^2 \Big ) \le C_{\tau ,T}. \end{aligned}$$\end{document}

This yields that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} is uniformly bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);H^3(B_R))$$\end{document} . Using the Gagliardo-Nirenberg inequality and the Sobolev embedding,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^4 \le C \Vert c_\varepsilon (t)\Vert _{H^{3}(B_R)}^2 \Vert \nabla c_\varepsilon (t)\Vert _{L^{4}(B_R)}^2 \le C \Vert c_\varepsilon (t)\Vert _{H^{3}(B_R)}^2 \Vert c_\varepsilon (t)\Vert _{H^{2}(B_R)}^2. \end{aligned}$$\end{document}

Subsequently, using the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,\infty );H^2(\Omega ))$$\end{document} again, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^T \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^4 \le C \int _0^T \Vert c_\varepsilon (t)\Vert _{H^{3}(B_R)}^2 \le C_{\tau ,T}. \end{aligned}$$\end{document}

Finally, by the boundedness (2.40), an application of the Grönwall inequality to (2.39) shows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{B_R} n_\varepsilon ^p(t) \le \left( \int _{B_R} n_0^p \right) \exp \left( \sup _{\varepsilon >0} \left( \int _0^T \Vert \nabla c_\varepsilon (t)\Vert _{L^{8}(B_R)}^4 \right) \right) \le C_{\tau ,T}, \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<t<T$$\end{document} . The gradient estimate in (2.36) is obtained by choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.12

It holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \Big ( \Vert n_\varepsilon \Vert _{L^\infty (B_R\times (0,T))} \Big ) \le C_{\tau ,T}, \end{aligned}$$\end{document}

and, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \Big ( \Vert w_\varepsilon \Vert _{L^\infty ((0,T);W^{1,\infty }(B_R))} + \Vert \Delta w_\varepsilon \Vert _{L^p(B_R\times (0,T))} + \Vert c_\varepsilon \Vert _{L^\infty ((0,T);W^{2,\infty }(B_R))} \Big ) \le C_{\tau ,T}. \end{aligned}$$\end{document}

Consequently, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \left( \Vert n_\varepsilon \Vert _{C^{\gamma ,\gamma /2}(\overline{B_R}\times [0,T])} \right) \le C_{\tau ,T} . \end{aligned}$$\end{document}

Proof

Using the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^p(B_R))$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p <\infty $$\end{document} , we can similar arguments to Lemma 2.5, with a suitable Hölder inequality to account for different regularities of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\varepsilon $$\end{document} in this case, and the estimate (2.12) to prove the estimate (2.41). Then, by repeating the techniques of Lemma 2.6 with the maximal regularity and the smoothing effect of the Neumann heat semigroup, we obtain (2.42), which allows us to derive (2.43) similarly to Lemma 2.7. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We are ready to prove the remaining case of Theorem 1.1.

Proof of Theorem 1.1 in critical dimension N=4

Based on the uniform regularity in Lemma 2.12, we can repeat all the steps and arguments in the proof for the subcritical case in Subsection 2.2.3. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Convergence rates and the initial layer’s effect for PES

In this section, we investigate the accuracy of the parabolic-elliptic simplification presented in Theorem 1.1, with the main result stated in Theorem 1.2. We begin with estimates for the initial layers in Lemma 3.1 and obtain needed regularity for the limiting solution in Lemma 3.2. Then, we present energy estimates for the rate system (1.15) in Lemmas 3.3-3.4, which help us prove Theorem 1.2. Recall that we consider all dimensions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le N\le 4$$\end{document} in this section.

Lemma 3.1

There exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\Vert \widetilde{c}_{\varepsilon }(0)\Vert _{W^{k+1,p}(\Omega )} \le } \,&\, {C \Vert -\Delta c_0 + c_0 -w_0\Vert _{W^{k,p}(\Omega )},} \\ {\Vert \widetilde{w}_{\varepsilon }(0)\Vert _{W^{l+1,p}(\Omega )} \le } \,&\, {C \Vert -\tau \Delta w_0 + w_0 - n_0\Vert _{W^{l,p}(\Omega )}}. \end{aligned} \end{aligned}$$\end{document}

Proof

The values c(0) and w(0) will be calculated from the equations for c and w in System (1.7)-(1.8), using the representations of the inverse operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta + I)^{-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\tau \Delta + I)^{-1}$$\end{document} , similarly to the proof of Lemma 2.8. Indeed, it follows from

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lll} c(x,t) \; = \displaystyle \int _0^{\infty } e^{s(\Delta - I)}w(x,t) ds, \\ w(x,t) = \displaystyle \int _0^{\infty } e^{s(\tau \Delta - I)}n(x,t) ds, \end{array}\right. \end{aligned}$$\end{document}

that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \widetilde{c}_{\varepsilon }(x,0) = \,&\, - \int _0^{\infty } e^{s(\Delta - I)}(\Delta - I)c_0(x) ds - \int _0^{\infty } e^{s(\Delta - I)}w_0(x) ds \nonumber \\ = \,&\, \int _0^{\infty } e^{s(\Delta - I)}[-\Delta c_0(x) + c_0(x) -w_0(x)] ds , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \widetilde{w}_{\varepsilon }(x,0) = \int _0^{\infty } e^{s(\tau \Delta - I)}[-\tau \Delta w_0(x) + w_0(x) - n_0(x)] ds. \end{aligned}$$\end{document}

Then, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p-L^q$$\end{document} estimates for the Neumann heat semigroup in Subsection A shows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \widetilde{c}_{\varepsilon }(0)\Vert _{W^{k+1,p}(\Omega )} \le C \left( \int _0^{\infty } e^{-s} s^{-\frac{1}{2}} ds \right) \Vert -\Delta c_0 + c_0 -w_0\Vert _{W^{k,p}(\Omega )}, \\&\Vert \widetilde{w}_{\varepsilon }(0)\Vert _{W^{l+1,p}(\Omega )} \le C \left( \int _0^{\infty } e^{-s} s^{-\frac{1}{2}} ds \right) \Vert -\tau \Delta w_0 + w_0 - n_0\Vert _{W^{l,p}(\Omega )}, \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k,l\in \mathbb N$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le p\le \infty $$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 3.2

Let (n, w, c) be the solution of System (1.7)-(1.8) as obtained in Theorem 1.1. Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \partial _{t}w \Vert _{L^\infty ((0,T);L^2(\Omega ))} + \Vert \partial _{t}w \Vert _{L^p(\Omega _T)} + \Vert \partial _{t}c \Vert _{L^\infty ((0,T);H^1(\Omega ))} + \Vert \partial _{t}c \Vert _{L^p(\Omega _T)} \le C_{T}, \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} .

Proof

Differentiating with respect to time from the representation (3.1) and using the equation for n from System (1.7)-(1.8), we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{t}w(t) = \displaystyle \int _0^{\infty } e^{s(\tau \Delta - I)} \big [ \Delta n(t) - \nabla n(t) \nabla c(t) - n(t) \Delta c(t) \big ] ds. \end{aligned}$$\end{document}

Moreover, we note from Theorem 1.1 and the standard regularisation that the solution (n, w, c) can be directly regularised to be sufficiently smooth, which allows us applying the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p-L^q$$\end{document} estimate for the heat Neumann semigroup, cf. (A.2) to obtain the desired estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t w$$\end{document} . The term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t c$$\end{document} is treated similarly using again the representation (3.1). On the other hand, the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{t} c $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial _{t} w $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(\Omega _T)$$\end{document} can be obtained directly via the maximal regularity. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le k\in \mathbb N$$\end{document} , based on the uniform regularity of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} -dependent solution given in Theorem 1.1, we will obtain an a priori estimate for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2k}$$\end{document} -energy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{n}_{\varepsilon }$$\end{document} due to direct computations from the rate system.

Lemma 3.3

For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 1$$\end{document} , there exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{k,T}>0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{d}{dt} \int _{\Omega }\widetilde{n}_{\varepsilon }^{2k}(t) \le -\dfrac{2k-1}{k} \int _{\Omega }|\nabla \widetilde{n}_{\varepsilon }^{k}|^2 + C_{k, T}\int _{\Omega }\widetilde{n}_{\varepsilon }^{2k} + C_{k, T}\int _{\Omega }|\nabla \widetilde{c}_{\varepsilon }|^2, \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<t<T$$\end{document} .

Proof

It is obvious from the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{n}_{\varepsilon }$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{d}{dt} \int _{\Omega }\widetilde{n}_{\varepsilon }^{2k}&=2k \int _{\Omega }\widetilde{n}_{\varepsilon }^{2k-1} \partial _{t} \widetilde{n}_{\varepsilon }= 2k \int _{\Omega }\widetilde{n}_{\varepsilon }^{2k-1} \left[ \Delta \widetilde{n}_{\varepsilon }- \nabla \cdot (\widetilde{n}_{\varepsilon }\nabla c_\varepsilon + n\nabla \widetilde{c}_{\varepsilon }) \right] \\&= -\dfrac{2(2k-1)}{k} \int _{\Omega }|\nabla \widetilde{n}_{\varepsilon }^{k}|^2 + {2(2k-1) \int _{\Omega }\widetilde{n}_{\varepsilon }^{k} \nabla \widetilde{n}_{\varepsilon }^{k} \cdot \nabla c_\varepsilon } \\&\hspace{0.45cm} + {2(2k-1) \int _{\Omega }n \widetilde{n}_{\varepsilon }^{k-1} \nabla \widetilde{n}_{\varepsilon }^{k} \cdot \nabla \widetilde{c}_{\varepsilon }} \\&\le -\dfrac{(2k-1)}{k} \int _\Omega |\nabla \widetilde{n}_{\varepsilon }^{k}|^2 + C_{k,T} \int _{\Omega }\widetilde{n}_{\varepsilon }^{2k} + C_{k,T} \int _\Omega |\nabla \widetilde{c}_{\varepsilon }|^{2} , \end{aligned}$$\end{document}

where we used the Young inequality and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert n\Vert _{L^{\infty }(\Omega _T)} + \Vert \widetilde{n}_{\varepsilon }\Vert _{L^\infty {(\Omega _T)}} \le C_T$$\end{document} at the last step. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The following lemma is obtained straightforwardly by testing the equations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{c}_{\varepsilon }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{w}_{\varepsilon }$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{c}_{\varepsilon }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{w}_{\varepsilon }$$\end{document} , and by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta ^2 \widetilde{c}_{\varepsilon }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta \widetilde{w}_{\varepsilon }$$\end{document} , respectively, then using integration by parts as well as Young’s inequality. Therefore, its proof is omitted.

Lemma 3.4

There hold that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\varepsilon \frac{d}{dt} \int _\Omega \widetilde{c}_{\varepsilon }^{\,2} + 2\int _\Omega |\nabla \widetilde{c}_{\varepsilon }|^2 + \int _\Omega \widetilde{c}_{\varepsilon }^{\,2} \le 2\int _\Omega \widetilde{w}_{\varepsilon }^2 + 2\varepsilon ^2 \int _\Omega |\partial _t c|^2, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\varepsilon \frac{d}{dt} \int _\Omega \widetilde{w}_{\varepsilon }^2 + 2 \tau \int _\Omega |\nabla \widetilde{w}_{\varepsilon }|^2 + \int _\Omega \widetilde{w}_{\varepsilon }^2 \le 2 \int _\Omega \widetilde{n}_{\varepsilon }^2 + 2 \varepsilon ^2 \int _\Omega |\partial _t w|^2, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\varepsilon \frac{d}{dt} \int _\Omega |\Delta \widetilde{c}_{\varepsilon }|^{2} + \int _\Omega |\nabla \Delta \widetilde{c}_{\varepsilon }|^{2} + 2\int _\Omega |\Delta \widetilde{c}_{\varepsilon }|^2 \le 2\int _\Omega |\nabla \widetilde{w}_{\varepsilon }|^2 + 2\varepsilon ^2 \int _\Omega |\nabla \partial _t c|^2, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\varepsilon \frac{d}{dt} \int _\Omega |\nabla \widetilde{w}_{\varepsilon }|^2 + \tau \int _\Omega |\Delta \widetilde{w}_{\varepsilon }|^2 + 2 \int _\Omega |\nabla \widetilde{w}_{\varepsilon }|^2 \le \frac{2}{\tau } \int _\Omega (\widetilde{n}_{\varepsilon })^2 + \frac{2\varepsilon ^2}{\tau } \int _\Omega |\partial _t w|^2. \end{aligned}$$\end{document}

We now prove Theorem 1.2.

Proof of Theorem 1.2

a) This part is proved by exploiting Lemma 3.3 together with Lemmas 3.1-3.2. Indeed, applying Lemma 3.3 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{d}{dt} \int _{\Omega }\widetilde{n}_{\varepsilon }^2 + \int _{\Omega }|\nabla \widetilde{n}_{\varepsilon }|^2 \le C_{ T}\int _{\Omega }\widetilde{n}_{\varepsilon }^2 + C_{ T}\int _{\Omega }|\nabla \widetilde{c}_{\varepsilon }|^2, \end{aligned}$$\end{document}

where we note that the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_T$$\end{document} does not depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} . A linear combination of the estimates in (3.8) and (3.4), (3.5) yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\frac{d}{dt} \int _\Omega \left[ \widetilde{n}_{\varepsilon }^2(t) + \varepsilon \left( \frac{C_T}{2} \widetilde{c}_{\varepsilon }^{\,2}(t) + C_T \widetilde{w}_{\varepsilon }^2(t) \right) \right] + \int _{\Omega } |\nabla \widetilde{n}_{\varepsilon }|^2 \\&\quad \le 3 C_T \int _{\Omega } \widetilde{n}_{\varepsilon }^2 + C_T \varepsilon ^2 \int _{\Omega } |\partial _t c|^2 + 2C_T \varepsilon ^2 \int _{\Omega } |\partial _t w|^2 , \end{aligned} \end{aligned}$$\end{document}

in which the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_T$$\end{document} is kept similarly to the first one. Taking into account the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t c, \partial _t w$$\end{document} given in Lemma 3.2, the last two terms on the right-hand side are bounded by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_T \varepsilon ^2$$\end{document} . Applying the Grönwall inequality, we obtain for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [0,T]$$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega \left[ \widetilde{n}_{\varepsilon }^2(t) + \varepsilon \left( \frac{C_T}{2} \widetilde{c}_{\varepsilon }^{\,2}(t) + C_T \widetilde{w}_{\varepsilon }^2(t) \right) \right] \le C_T \left[ \varepsilon ^2 + \varepsilon \int _\Omega \left( \frac{C_T}{2} \widetilde{c}_{\varepsilon }^{\,2}(0) + C_T \widetilde{w}_{\varepsilon }^2(0) \right) \right] , \end{aligned}$$\end{document}

where we note from the initial condition (1.17) that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{n}_{\varepsilon }(0)=0$$\end{document} . Thanks to Lemma 3.1,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega \left( \frac{C_T}{2} \widetilde{c}_{\varepsilon }^{\,2}(0) + C_T \widetilde{w}_{\varepsilon }^2(0) \right) \le \,&\, C \left( \Vert -\Delta c_0 + c_0 -w_0\Vert _{L^2(\Omega )}^2 + \Vert -\tau \Delta w_0 + w_0 - n_0\Vert _{L^2(\Omega )}^2 \right) \\ = \,&\, {C \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2}. \end{aligned}$$\end{document}

Therefore, the rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{n}_{\varepsilon }$$\end{document} , considered in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^2(\Omega ))$$\end{document} , is of the order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon + \sqrt{\varepsilon } \textrm{dist} [(n_0, c_0,w_0);\mathcal C_{\textsf{PES}}] )$$\end{document} , which consequently shows the first part of (1.22). The second part follows from integrating (3.9) over the time interval (0, t) and using the first part. For estimating the rate component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{w}_{\varepsilon }$$\end{document} , using the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t w$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^2(\Omega ))$$\end{document} in Lemma 3.2, and the rate estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{n}_{\varepsilon }$$\end{document} as (1.22), we have from (3.7) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varepsilon \frac{d}{dt} \int _\Omega |\nabla \widetilde{w}_{\varepsilon }|^2 + \tau \int _\Omega |\Delta \widetilde{w}_{\varepsilon }|^2 + 2 \int _\Omega |\nabla \widetilde{w}_{\varepsilon }|^2&{\le \frac{2}{\tau } \int _\Omega (\widetilde{n}_{\varepsilon })^2 + \frac{2\varepsilon ^2}{\tau } \int _\Omega |\partial _t w|^2}\\&{\le \frac{C_T}{\tau } \left( \varepsilon ^2 + \varepsilon \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) }, \end{aligned}$$\end{document}

Hence, by Lemma A.6, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega |\nabla \widetilde{w}_{\varepsilon }(t)|^2&\le e^{-\frac{2}{\varepsilon }t} \int _\Omega |\nabla \widetilde{w}_{\varepsilon }(0)|^2 + C_{\tau ,T} \left( \varepsilon + \big ( \textrm{dist}[(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) \int _0^t e^{-\frac{2}{\varepsilon }s} ds \\&\le e^{-\frac{2}{\varepsilon }t} \Vert \nabla \widetilde{w}_{\varepsilon }(0)\Vert _{L^2(\Omega )}^2 + C_{\tau ,T}\left( \varepsilon ^2 + \varepsilon \big ( \textrm{dist}[(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) \\&\le C {\Vert -\tau \Delta w_0 + w_0 - n_0\Vert _{H^1(\Omega )}^2} + C_{\tau ,T}\left( \varepsilon ^2 + \varepsilon \big ( \textrm{dist}[(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) , \end{aligned}$$\end{document}

where we note that the distances on the right-hand side are less than or equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}^{0,1}_2[(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]$$\end{document} . We derive the estimate (1.23), where the zeroth order term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _\Omega |\widetilde{w}_{\varepsilon }(t)|^2$$\end{document} is estimated in the same way. The proof of (1.24) follows similarly, so we omit it here.

b) It is sufficient to prove this part for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2k$$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 1$$\end{document} ). Thanks to the rate estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{n}_{\varepsilon }$$\end{document} in Part a, the estimate (3.5), and Lemma 3.2, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _t} \widetilde{w}_{\varepsilon }^2&\le \varepsilon \int _{\Omega } \widetilde{w}_{\varepsilon }^2(0) + C \left( \iint _{\Omega _t} \widetilde{n}_{\varepsilon }^2 + \varepsilon ^2 \iint _{\Omega _t} |\partial _s w|^2 \right) \\&\le \varepsilon {\Vert -\tau \Delta w_0 + w_0 - n_0\Vert _{L^2(\Omega )}^2} + C_T \left( \varepsilon ^2 + \varepsilon \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) \\&\le C_T \left( \varepsilon ^2 + \varepsilon \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) . \end{aligned}$$\end{document}

Then, by integrating (3.4) on (0, t), we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _t} |\nabla \widetilde{c}_{\varepsilon }|^2&\le C_T \left( \varepsilon ^2 + \varepsilon \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) . \end{aligned}$$\end{document}

Now, it follows from Lemma 3.3 that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega }\widetilde{n}_{\varepsilon }^{p}(t)&\le \int _{\Omega }\widetilde{n}_{\varepsilon }^{p}(0) -\dfrac{2p-2}{p} \iint _{\Omega _t} |\nabla \widetilde{n}_{\varepsilon }^{p/2}|^2 + C_{p, T} \iint _{\Omega _t} \widetilde{n}_{\varepsilon }^{p} + C_{p, T} \iint _{\Omega _t} |\nabla \widetilde{c}_{\varepsilon }|^2 \\&\le \int _{\Omega }\widetilde{n}_{\varepsilon }^{p}(0) + C_{p, T} \iint _{\Omega _t} \widetilde{n}_{\varepsilon }^{p} + C_{p, T,\tau } \left( \varepsilon ^2 + \varepsilon \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}] \big )^2 \right) . \end{aligned}$$\end{document}

The Grönwall inequality directly shows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{n}_{\varepsilon }\Vert _{L^{\infty }((0,T); L^{p}(\Omega ))} \le C_{p,T,\tau } \left( \varepsilon ^{\frac{2}{p}} + \varepsilon ^{\frac{1}{p}} \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]\big )^{\frac{2}{p}} \right) . \end{aligned}$$\end{document}

Thanks to the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{t} c $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial _{t} w $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q(\Omega _T)$$\end{document} in Lemma 3.2 and the estimate (3.10), we apply the maximal regularity with slow evolution (cf. Lemma A.4) to the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{w}_{\varepsilon }$$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert \widetilde{w}_{\varepsilon }\Vert _{L^p((0,T);W^{2,p}(\Omega ))} \le C_{p,\tau } \left( \varepsilon ^{\frac{1}{p}} \Vert \Delta \widetilde{w}_{\varepsilon }(0) \Vert _{L^p(\Omega )} + \Vert \widetilde{n}_{\varepsilon }- \varepsilon \partial _{t} w \Vert _{L^p(\Omega _T)} \right) \\&\quad \le C_{p,\tau ,T} \left( \varepsilon ^{\frac{1}{p}} {\Vert -\tau \Delta w_0 + w_0 - n_0\Vert _{W^{2,p}(\Omega )}} + \varepsilon ^{\frac{2}{p}} + \varepsilon ^{\frac{1}{p}} \big ( \textrm{dist} [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]\big )^{\frac{2}{p}} \right) \\&\quad \le C_{p,\tau ,T} \left( \varepsilon ^{\frac{2}{p}} + \varepsilon ^{\frac{1}{p}} \big ( \textrm{dist}^{0,2}_p [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]\big )^{\frac{2}{p}} \right) . \end{aligned}$$\end{document}

Similarly, we have the following estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \widetilde{c}_{\varepsilon }\Vert _{L^p((0,T);W^{4,p}(\Omega ))}&\le C_{p} \left( \varepsilon ^{\frac{1}{p}} \Vert \Delta ^2 \widetilde{c}_{\varepsilon }(0) \Vert _{L^p(\Omega )} + \Vert \Delta \widetilde{w}_{\varepsilon }- \varepsilon \partial _{t} \Delta c \Vert _{L^p(\Omega _T)} \right) \\&\le C_{p,\tau ,T} \left( \varepsilon ^{\frac{2}{p}} + \varepsilon ^{\frac{1}{p}} \big ( \textrm{dist}^{4,2}_p [(n_0,c_0,w_0);\mathcal C_{\textsf{PES}}]\big )^{\frac{2}{p}} \right) , \end{aligned}$$\end{document}

which completes the proof.

From indirect signalling to direct signalling

We rigorously study the indirect-direct simplification from (1.1)-(1.3) to (1.29) in this section. The main result of this part was stated in Theorem 1.3, including both passing to the limit and estimating the convergence rates.

Balancing the multiple time scale Lyapunov functional

Lemma 4.1

It holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{t>0}\int _\Omega \left( n_\kappa \log n_\kappa + \frac{1}{4}|\Delta c_\kappa - c_\kappa + w_\kappa |^2 + \frac{1}{4} |\nabla c_\kappa |^2 + \frac{1}{2} c_\kappa ^2 \right) \nonumber \\&\quad + \frac{1}{\varepsilon } \iint _{\Omega _\infty } \Big ( |\nabla (\Delta c_\kappa - c_\kappa + w_\kappa )|^2 + |\Delta c_\kappa - c_\kappa + w_\kappa |^2 \Big ) \le C . \end{aligned}$$\end{document}

Proof

Similarly to the prooof of Lemma 2.9, we will balance the dissipation inequality in Lemma 2.2. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=1$$\end{document} then the Sobolev embedding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(\Omega ) \hookrightarrow L^\infty (\Omega )$$\end{document} can be utilised to see that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega } n_\kappa c_\kappa \le \Vert c_\kappa \Vert _{L^\infty (\Omega )} \int _{\Omega } n_\kappa \le C M \Vert c_\kappa \Vert _{H^1(\Omega )} \le \frac{1}{4} \Vert c_\kappa \Vert _{H^1(\Omega )}^2 + C^2M^2. \end{aligned}$$\end{document}

By skipping the term including \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} on the left-hand side of (2.9), for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _\Omega \left( n_\kappa \log n_\kappa + \frac{1}{2}|\Delta c_\kappa - c_\kappa + w_\kappa |^2 + \frac{1}{2} |\nabla c_\kappa |^2 + \frac{1}{2} c_\kappa ^2 \right) \\&\qquad + \frac{1}{\varepsilon } \iint _{\Omega _t} \Big ( |\nabla (\Delta c_\kappa - c_\kappa + w_\kappa )|^2 + |\Delta c_\kappa - c_\kappa + w_\kappa |^2 \Big ) \\&\quad \le \mathcal E(n_0,c_0) + \int _\Omega n_\kappa c_\kappa \le \mathcal E(n_0,c_0) + \frac{1}{4} \Vert c_\kappa \Vert _{H^1(\Omega )}^2 + C . \end{aligned}$$\end{document}

The estimate (4.1) is showed by absorbing the term including \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert c_\kappa \Vert _{H^1(\Omega )}$$\end{document} to the left-hand side.

Let us consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} by exploiting the Moser-Trudinger inequality (instead of the Adam type inequality), which is represented in Part a of Lemma A.3. Indeed, for any positive real number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} to be chosen later, a combination of the inequalities (A.5) and (A.6) gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega } n_\kappa c_\kappa&\le \frac{1}{e} + \frac{1}{\alpha } \int _\Omega n_\kappa \log n_\kappa + \frac{\Vert n_\kappa \Vert _{L^1(\Omega )}}{\alpha } \log \left( \int _{\Omega }e^{\,\alpha \, c_\kappa } \right) \\&\le \frac{1}{e} + \frac{1}{\alpha } \int _{\Omega }n_\kappa \log n_\kappa + \frac{M}{\alpha } \left[ \frac{\alpha ^2}{8\pi } \Vert \nabla c_\kappa \Vert _{L^2(\Omega )}^2 + \frac{\alpha }{|\Omega |} \int _\Omega c_\kappa + C_{\alpha } \right] \\&= \frac{1}{\alpha } \int _{\Omega }n_\kappa \log n_\kappa + \frac{\alpha M}{8\pi } \int _{\Omega }|\nabla c_\kappa |^2 + C_{\alpha ,M,\Omega }. \end{aligned}$$\end{document}

Consequently, it follows from (2.9) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\int _{\Omega }\left( n_\kappa \log n_\kappa + \frac{1}{2}|\Delta c_\kappa - c_\kappa + w_\kappa |^2 + \frac{1}{2} |\nabla c_\kappa |^2 + \frac{1}{2} c_\kappa ^2 \right) \\&\qquad + \frac{1}{\varepsilon } \iint _{\Omega _t} \Big ( |\nabla (\Delta c_\kappa - c_\kappa + w_\kappa )|^2 + |\Delta c_\kappa - c_\kappa + w_\kappa |^2 \Big ) \\&\quad \le \mathcal E(n_0,c_0) + \frac{1}{\alpha } \int _{\Omega }n_\kappa \log n_\kappa + \frac{\alpha M}{8\pi } \int _{\Omega }|\nabla c_\kappa |^2 + C . \end{aligned} \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M<4\pi $$\end{document} , there always exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha M/(8\pi ) < 1/2$$\end{document} , which means that the integrals on the latter right-hand side can be controlled by terms on the left-hand side. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.2

It holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \,\in (0,\infty )^2} \Big ( \Vert n_\kappa \Vert _{L^2(\Omega _T)} + \Vert c_\kappa \Vert _{L^2((0,T);H^2(\Omega ))} \Big ) \le C. \end{aligned}$$\end{document}

Proof

By considering the Boltzmann entropy for the first equation of (1.1), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega (n_\kappa \log n_\kappa +e^{-1})&= \int _\Omega (n_{0} \log n_{0} +e^{-1}) - {\iint _{\Omega _t}} \frac{|\nabla n_\kappa |^2}{n_\kappa } - {\iint _{\Omega _t}} n_\kappa \Delta c_\kappa \\&\le C - \iint _{\Omega _{T}}\frac{|\nabla n_\kappa |^2}{n_\kappa } + \frac{1}{2} \iint _{\Omega _{T}}n_\kappa ^2+ \frac{1}{2} \iint _{\Omega _{T}}|\Delta c_\kappa |^2 , \end{aligned}$$\end{document}

which shows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega (n_\kappa \log n_\kappa +e^{-1}) + \iint _{\Omega _{T}}\frac{|\nabla n_\kappa |^2}{n_\kappa } \le C + \frac{1}{2} \iint _{\Omega _{T}}n_\kappa ^2+ \frac{1}{2} \iint _{\Omega _{T}}|\Delta c_\kappa |^2. \end{aligned}$$\end{document}

We will balance the two sides of the above estimate. Thanks to the parabolic maximal regularity with slow evolution (see Lemma A.4) applied to the second equation of System (1.1),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert \Delta c_\kappa \Vert _{L^2(\Omega _T)}&\le C \left( \varepsilon ^{\frac{1}{2}} \Vert \Delta c_0\Vert _{L^2(\Omega )} + \Vert w_\kappa \Vert _{L^2(\Omega _T)} \right) \\&\le C \left( \varepsilon ^{\frac{1}{2}} \Vert \Delta c_0\Vert _{L^2(\Omega )} + {\left( \varepsilon \int _{\Omega }w_0^2 + \iint _{\Omega _{T}}n_\kappa ^2\right) ^{1/2}} \right) , \end{aligned} \end{aligned}$$\end{document}

where the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega _T)$$\end{document} -norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\kappa $$\end{document} is controlled by testing the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\kappa $$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\kappa $$\end{document} , given as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\varepsilon }{2} \int _{\Omega }w_\kappa ^2 + \tau \iint _{\Omega _{T}}|\nabla w_\kappa |^2 + \frac{1}{2}\iint _{\Omega _{T}}w_\kappa ^2 \le \frac{\varepsilon }{2} \int _{\Omega }w_0^2 + \frac{1}{2} \iint _{\Omega _{T}}n_\kappa ^2. \end{aligned}$$\end{document}

Therefore, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega (n_\kappa \log n_\kappa +e^{-1}) + \iint _{\Omega _{T}}\frac{|\nabla n_\kappa |^2}{n_\kappa } \le C + C \iint _{\Omega _{T}}n_\kappa ^2 . \end{aligned}$$\end{document}

Due to Lemma 4.1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,\infty );L\log L(\Omega ))$$\end{document} , which suits to apply Lemma A.1 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\le 2$$\end{document} to have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _{T}}n_\kappa ^2&\le \alpha \left( \sup _{t>0} \int _{\Omega } (n_\kappa \log n_\kappa + e^{-1}) \right) \iint _{\Omega _{T}}\frac{|\nabla n_\kappa |^2}{n_\kappa } + C_\alpha T , \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} . Consequently,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _{T}}n_\kappa ^2&\le C \alpha \iint _{\Omega _{T}}\frac{|\nabla n_\kappa |^2}{n_\kappa } + C_\alpha T. \end{aligned}$$\end{document}

This estimate yields that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega _T)$$\end{document} -norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} in (4.3) can be controlled by the second term on the left-hand side with a sufficiently small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} . Hence, we obtain the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla n_\kappa |^2/n_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1(\Omega _T)$$\end{document} , which in a combination with (4.4) concludes that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega _T)$$\end{document} . Then, back to (4.2), we obtain a uniform bound for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);H^2(\Omega ))$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.3

There is a positive constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \text {ess sup}_{t\in (0,T)} \int _{\Omega }n_\kappa ^{1+\frac{\delta }{2}}(t) + \Vert n_\kappa \Vert _{L^{2+\delta }(\Omega _T)} + \iint _{\Omega _T} n_\kappa ^{\frac{\delta }{2}-1} |\nabla n_\kappa |^2 \right) \le C_{T}. \end{aligned}$$\end{document}

Proof

Considering the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} -energy functional (with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document} ) for the first component of (1.1),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \frac{1}{p} \int _\Omega n_\kappa ^p(t)&= -(p-1) \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + \frac{1}{p} \int _\Omega n_{0}^p + (p-1) \iint _{\Omega _{t}}n_\kappa ^{p-1}\nabla n_\kappa \cdot \nabla c_\kappa \\&= -(p-1) \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + \frac{1}{p} \int _\Omega n_{0}^p - \frac{p-1}{p} \iint _{\Omega _{t}}n_\kappa ^{p} \Delta c_\kappa \\&\le -(p-1) \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + \frac{1}{p} \int _\Omega n_{0}^p + \underbrace{\frac{p-1}{p} \Vert n_\kappa \Vert _{L^{2p}(\Omega _t)}^p \Vert \Delta c_\kappa \Vert _{L^{2}(\Omega _t)}}_{:=J_\kappa (t)} , \end{aligned} \end{aligned}$$\end{document}

in which, we note from Lemma 4.2 that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta c_\kappa $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega _T)$$\end{document} . To balance the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} -energy functional, we will estimate the temporal supremum of the integral on the left-hand side, or more precisely, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^p(t) $$\end{document} . Using the Gagliardo-Nirenberg inequality of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert f\Vert _{L^4(\Omega )}^4 \le C \Vert \nabla f\Vert _{L^2(\Omega )}^2 \Vert f\Vert _{L^2(\Omega )}^2 + C \Vert f\Vert _{L^2(\Omega )}^4 \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=n_\kappa ^{p/2}(s)$$\end{document} (here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0,t)$$\end{document} ), we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _\Omega n_\kappa ^{2p}(s)&\le Cp^2 \int _\Omega n_\kappa ^{p-2}(s) |\nabla n_\kappa (s)|^2 \int _\Omega n_\kappa ^p(s) + C\int _\Omega n_\kappa ^p(s) . \end{aligned}$$\end{document}

Thus, we can estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \iint _{\Omega _{t}}n_\kappa ^{2p}\right) ^{\frac{1}{2}}&\le \left( Cp^2 \, \text {ess sup}_{s\in (0,T)} \int _\Omega n_\kappa ^p(s) \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + C t \text {ess sup}_{s\in (0,T)} \int _\Omega n_\kappa ^p(s) \right) ^{\frac{1}{2}}\nonumber \\&= Cp \left( \text {ess sup}_{s\in (0,T)} \int _\Omega n_\kappa ^p(s) \right) ^{\frac{1}{2}} \left( \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + \frac{C}{p^2} t \right) ^{\frac{1}{2}} . \end{aligned}$$\end{document}

Then, the last term of the energy estimate (4.6) can be treated as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J_\kappa (t)&\le C(p-1) \Vert \Delta c_\kappa \Vert _{L^{2}(\Omega _T)} \left( \text {ess sup}_{s\in (0,T)} \int _\Omega n_\kappa ^p(s) \right) ^{\frac{1}{2}} \left( \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + \frac{CT}{p^2} \right) ^{\frac{1}{2}} \\&\le C_T(p-1) \left( \text {ess sup}_{s\in (0,T)} \int _\Omega n_\kappa ^p(s) \right) ^{\frac{1}{2}} \left( \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + \frac{CT}{p^2} \right) ^{\frac{1}{2}} \\&\le \frac{p-1}{2} \left( \iint _{\Omega _{t}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 + \frac{CT}{p^2} \right) + C_{T}(p-1) \left( \text {ess sup}_{s\in (0,T)} \int _\Omega n_\kappa ^p(s) \right) , \end{aligned}$$\end{document}

using the Young inequality. Combining this with the energy estimate (4.6) (and replacing the variable s by t in the above supremum), we derive

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{1}{p} \int _\Omega n_\kappa ^p(t) + \frac{p-1}{2} \iint _{\Omega _{T}}n_\kappa ^{p-2} |\nabla n_\kappa |^2 \le C_{T,\,p} + (p-1)C_{T} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^p(t) \right) , \end{aligned}$$\end{document}

where it is useful to note that the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_T$$\end{document} does not depend on p. Subsequently, by skipping the gradient term and then taking the supremum over time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in (0,T)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{p} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^p(t) \right) \le C_{T,\,p} + C_{T}(p-1) \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^p(t) \right) . \end{aligned}$$\end{document}

Choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1+\delta /2$$\end{document} in which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} is sufficiently small such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{T}p(p-1)<1$$\end{document} , we obtain the uniform boundedness

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^p(t) \right) \le \frac{p C_{T,\,p}}{1-C_{T}p(p-1)} , \end{aligned}$$\end{document}

i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);L^{1+\delta /2}(\Omega ))$$\end{document} . Then, we obtain the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa ^{p-2} |\nabla n_\kappa |^2$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1(\Omega _T)$$\end{document} by returning to (4.9), and so is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2p}(\Omega _T) \equiv L^{2+\delta }(\Omega _T)$$\end{document} due to (4.8). The desired estimate (4.5) is proved. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} be the constant obtained in Lemma 4.3, and define the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p_j\}_{j=1,2,\dots }$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0>1 \quad \text {and} \quad p_{j+1}:=p_j+\delta /2 \text { for } j=0,1,\dots $$\end{document}

If for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\ge 0$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \Vert n_\kappa \Vert _{L^{2p_j}(\Omega _T)} \right) \le C_T, \end{aligned}$$\end{document}

then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) + \Vert n_\kappa \Vert _{L^{2p_{j+1}}(\Omega _T)} \right) \le C_{T,p_{j+1}}. \end{aligned}$$\end{document}

Proof

The main idea of this proof is to balance the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p_{j+1}}$$\end{document} -energy estimate from the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2p_j}(\Omega _T)$$\end{document} -regularity given in the assumption (4.10). Taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=p_{j+1}$$\end{document} in the energy estimate (4.6), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\frac{1}{p_{j+1}} \int _\Omega n_\kappa ^{p_{j+1}}(t) \\&\quad = -(p_{j+1}-1) \iint _{\Omega _{t}}n_\kappa ^{p_{j+1}-2} |\nabla n_\kappa |^2 + \frac{1}{p_{j+1}} \int _\Omega n_{0}^{p_{j+1}} - \frac{p_{j+1}-1}{p_{j+1}} \iint _{\Omega _{t}}n_\kappa ^{p_{j+1}} \Delta c_\kappa . \end{aligned} \end{aligned}$$\end{document}

To balance this energy, we also recall from a similar application of the Gagliardo-Nirenberg inequality (4.7) as the proof of Lemma 4.3 that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert n_\kappa \Vert _{L^{2p_{j+1}}(\Omega _T)}^{p_{j+1}}&\le Cp_{j+1} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) \right) ^{\frac{1}{2}} \left( \iint _{\Omega _{T}}n_\kappa ^{p_{j+1}-2}(s) |\nabla n_\kappa (s)|^2 + CT \right) ^{\frac{1}{2}} \\&\le C_{p_{j+1}} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) + \iint _{\Omega _{T}}n_\kappa ^{p_{j+1}-2}(s) |\nabla n_\kappa (s)|^2 + CT \right) . \end{aligned} \end{aligned}$$\end{document}

First, let us show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta c_\kappa $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2p_j}(\Omega _T)$$\end{document} , based on the assumption (4.10). Indeed, the parabolic maximal regularity with slow evolution (see Lemma A.4) yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert \Delta c_\kappa \Vert _{L^{2p_j}(\Omega _T)}&\le \left( \frac{\varepsilon }{2p_j} \right) ^{\frac{1}{2p_j}} \Vert \Delta c_0\Vert _{L^{2p_j}(\Omega )} + C_{p_j} \Vert w_\kappa \Vert _{L^{2p_j}(\Omega _T)} \\&\le C_{p_j} \left( \Vert \Delta c_0\Vert _{L^{2p_j}(\Omega )} + \Vert w_\kappa \Vert _{L^{2p_j}(\Omega _T)} \right) . \end{aligned} \end{aligned}$$\end{document}

Here, by using the third equation of (1.1),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w_\kappa \Vert _{L^{2p_j}(\Omega _T)}^{2p_j}&= \frac{\varepsilon }{2p_j} \int _\Omega (w_0^{2p_j}-w_\kappa ^{2p_j}) - \tau (2p_j-1) \iint _{\Omega _{T}}w_\kappa ^{2p_j-2} |\nabla w_\kappa |^2\\&\quad + \iint _{\Omega _{T}}n_\kappa w_\kappa ^{2p_j-1}. \end{aligned}$$\end{document}

Skipping the negative terms on the right-hand side, and applying the Young inequality as follows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n_\kappa w_\kappa ^{2p_j-1}\le \frac{1}{2p_j} n_\kappa ^{2p_j} + \frac{2p_j-1}{2p_j} w_\kappa ^{2p_j}, $$\end{document}

we obtain the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w_\kappa \Vert _{L^{2p_j}(\Omega _T)}^{2p_j} \le \varepsilon \int _\Omega w_0^{2p_j} + \Vert n_\kappa \Vert _{L^{2p_j}(\Omega _T)}^{2p_j} \le \int _\Omega w_0^{2p_j} + \Vert n_\kappa \Vert _{L^{2p_j}(\Omega _T)}^{2p_j} . \end{aligned}$$\end{document}

Therefore, we imply from (4.14) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta c_\kappa \Vert _{L^{2p_j}(\Omega _T)}&\le C_{p_j} \left( \Vert \Delta c_0\Vert _{L^{2p_j}(\Omega )} + \left( \int _\Omega w_0^{2p_j} + \Vert n_\kappa \Vert _{L^{2p_j}(\Omega _T)}^{2p_j} \right) ^{\frac{1}{2p_j}} \right) \\&\le C_{p_j} \left( \Vert \Delta c_0\Vert _{L^{2p_j}(\Omega )} + \Vert w_0 \Vert _{L^{2p_j}(\Omega )} + \Vert n_\kappa \Vert _{L^{2p_j}(\Omega _T)} \right) , \end{aligned}$$\end{document}

i.e., the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta c_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2p_j}(\Omega _T)$$\end{document} has been showed.

Now, we can estimate the term including \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa ^{p_{j+1}} \Delta c_\kappa $$\end{document} in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p_{j+1}}$$\end{document} -energy computation as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \iint _{\Omega _{t}}n_\kappa ^{p_{j+1}} \Delta c_\kappa \le \Vert n_\kappa \Vert _{L^{\frac{2p_{j}p_{j+1}}{2p_{j}-1}}(\Omega _T)}^{p_{j+1}} \Vert \Delta c_\kappa \Vert _{L^{2p_j}(\Omega _T)} \le C_{T,p_j} \Vert n_\kappa \Vert _{L^{\frac{2p_{j}p_{j+1}}{2p_{j}-1}}(\Omega _T)}^{p_{j+1}} . \end{aligned}$$\end{document}

By interpolation in Lebesgue spaces,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert n_\kappa \Vert _{L^{\frac{2p_{j}p_{j+1}}{2p_{j}-1}}(\Omega _T)}^{p_{j+1}}&\le \left( \Vert n_\kappa \Vert _{L^{2p_{j+1}}(\Omega _T)}^{\lambda _{j+1}} \Vert n_\kappa \Vert _{L^{1}(\Omega _T)}^{1-\lambda _{j+1}} \right) ^{p_{j+1}} \le C_{p_{j+1}}\Vert n_\kappa \Vert _{L^{2p_{j+1}}(\Omega _T)}^{\lambda _{j+1}p_{j+1}} , \end{aligned}$$\end{document}

where, by direct computation,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{j+1}=\frac{2p_{j}p_{j+1}-2p_{j}+1}{2p_{j}p_{j+1}-p_{j}} \in (0,1). \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{j+1}p_{j+1}<p_{j+1}$$\end{document} , for any constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta >0$$\end{document} the Young inequality ensures

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&- \iint _{\Omega _{t}}n_\kappa ^{p_{j+1}} \Delta c_\kappa \le C_{T,\,p_{j+1},\eta } + \eta \Vert n_\kappa \Vert _{L^{2p_{j+1}}(\Omega _T)}^{p_{j+1}} \\&\quad \le C_{T,\,p_{j+1},\eta } + \eta C_{p_{j+1}} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) + \iint _{\Omega _{T}}n_\kappa ^{p_{j+1}-2}(s) |\nabla n_\kappa (s)|^2 + CT \right) , \end{aligned}$$\end{document}

where we have used (4.13) at the second estimate. This combines with (4.12) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{1}{p_{j+1}} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) \right) + (p_{j+1}-1) \iint _{\Omega _{T}}n_\kappa ^{p_{j+1}-2} |\nabla n_\kappa |^2 \\&\quad \le C_{T,\,p_{j+1},\eta } + \eta C_{p_{j+1}} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) + \iint _{\Omega _{T}}n_\kappa ^{p_{j+1}-2}(s) |\nabla n_\kappa (s)|^2 + CT \right) . \end{aligned}$$\end{document}

By choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} sufficiently small, we can absorb the integrals on the right-hand side into the left one, which accordingly gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p_{j+1}}(t) + \iint _{\Omega _{T}}n_\kappa ^{p_{j+1}-2} |\nabla n_\kappa |^2 \le C_{T,\,p_{j+1}}. \end{aligned}$$\end{document}

With this boundedness, we finally obtain (4.11) using (4.13). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.5

It holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \Big ( \Vert n_\kappa \Vert _{L^\infty (\Omega _T)} + \Vert n_\kappa \Vert _{L^2((0,T);H^1(\Omega ))} \Big ) \le C_{T}, \end{aligned}$$\end{document}

and, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{\kappa \in (0,\infty )^2} \Big ( \Vert w_\kappa \Vert _{L^\infty (\Omega _T)} + \Vert w_\kappa \Vert _{L^2((0,T);H^{1}(\Omega ))} \Big ) \le C_{T}, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{\kappa \in (0,\infty )^2} \Big ( \Vert c_\kappa \Vert _{L^\infty ((0,T);W^{1,\infty }(\Omega ))} + \Vert c_\kappa \Vert _{L^p((0,T);W^{2,p}(\Omega ))} \Big ) \le C_{T}. \end{aligned}$$\end{document}

Consequently, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \Vert n_\kappa \Vert _{C^{\gamma ,\gamma /2}(\overline{\Omega }\times [0,T])} \right) \le C_{T} . \end{aligned}$$\end{document}

Proof

From Lemma 4.4, we obtain for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<\infty $$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\kappa \in (0,\infty )^2} \left( \text {ess sup}_{t\in (0,T)} \int _\Omega n_\kappa ^{p}(t) + \Vert n_\kappa \Vert _{L^{p}(\Omega _T)} + \iint _{\Omega _{T}}|\nabla n_\kappa |^2 \right) \le C_{T,p}, \end{aligned}$$\end{document}

where we note that the limit of (4.11) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\rightarrow \infty $$\end{document} has not been claimed because of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{j+1}$$\end{document} -dependence (i.e., the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} -boundedness is not a consequence of (4.11)). This implies (4.15) similarly to Lemma 2.5, noting again that we exploit the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (0,T;L^p(\Omega ))$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 1$$\end{document} , the estimate (2.12).

Now, it follows from the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _{T}}w_\kappa ^p&= - \frac{\varepsilon }{p} \int _{\Omega }w_\kappa ^p - \tau (p-1) \iint _{\Omega _{T}}w_\kappa ^{p-2} |\nabla w_\kappa |^2 + \frac{\varepsilon }{p} \int _{\Omega }w_0^p + \iint _{\Omega _{T}}n_\kappa w_\kappa ^{p-1} , \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document} . Then, by the Young inequality, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _{T}}w_\kappa ^p \le \frac{\varepsilon }{p} \int _{\Omega }w_0^p + \frac{1}{p} \iint _{\Omega _{T}}n_\kappa ^p + \frac{p-1}{p} \iint _{\Omega _{T}}w_\kappa ^p, \end{aligned}$$\end{document}

which consequently deduces that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \lim _{p\rightarrow \infty } \Vert w_\kappa \Vert _{L^p(\Omega _T)} \le \lim _{p\rightarrow \infty } \left( \varepsilon \Vert w_0\Vert _{L^p(\Omega )}^p + \Vert n_\kappa \Vert _{L^p(\Omega _T)}^p \right) ^{1/p}\\ & \quad \le C \left( \Vert w_0\Vert _{L^\infty (\Omega )} + \Vert n_\kappa \Vert _{L^\infty (\Omega _T)} \right) , \end{aligned}$$\end{document}

i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\kappa $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} . Based on the boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla n_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega _T)$$\end{document} , we can similarly test the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\kappa $$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta w_\kappa $$\end{document} to obtain the same boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla w_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega _T)$$\end{document} , and so is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);H^1(\Omega ))$$\end{document} as (4.16).

For the component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\kappa $$\end{document} , a uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bound in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty ((0,T);H^1(\Omega ))$$\end{document} was obtained in the Lemma 4.1. The first term in the estimate (4.17) is proved similarly to Lemma 2.6, while the second one is directly a consequence of the maximal regularity with slow evolution given in Lemma A.4. Finally, one can show (4.18) similarly to Lemma 2.7. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Passage to the limit and convergence rate analysis

Proof of Theorem 1.3

Based on the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} regularity obtained in Lemma 4.5, one can adapt all steps from the proof of Theorem 1.1 to prove the passage to the limit given in this theorem. For the convergence rate estimates, the estimate (3.9) still holds, i.e.,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{d}{dt} \int _\Omega \left[ (n_\kappa (t)-n(t))^2 + \varepsilon \left( \frac{C_T}{2} (c_\kappa (t)-c(t))^{\,2} + C_T (w_\kappa (t)-w(t))^2 \right) \right] \\&\qquad + \int _{\Omega } |\nabla (n_\kappa (t)-n(t))|^2 \\&\quad \le 3 C_T \int _{\Omega } (n_\kappa (t)-n(t))^2 + C_T \varepsilon ^2 \int _{\Omega } |\partial _t c|^2 + 2C_T \varepsilon ^2 \int _{\Omega } |\partial _t w|^2 , \end{aligned}$$\end{document}

since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla c_\kappa $$\end{document} is uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)^N$$\end{document} as Lemma 4.5, which consequently shows (1.34). On the other hand, by skipping the term including \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} at the estimate (3.5), and then using the comparison principle for differential equations in Lemma A.6, we get (1.35). The estimate (1.36) is obtained similarly to the rest of the proof of Theorem 1.2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

It remains to prove Corollary 2.

Proof of Corollary 2

The estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Delta c_\kappa - c_\kappa + w_\kappa \Vert _{L^2(0,T;H^1(\Omega ))} \le C\sqrt{|\kappa |} \end{aligned}$$\end{document}

follows immediately from (4.1). For the remaining part, we use the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} to write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (n_\kappa - w_\kappa )^2 = (n_\kappa - w_\kappa )(\varepsilon \partial _t w_\kappa - \tau \Delta w_\kappa ). \end{aligned}$$\end{document}

Therefore, straightforward computations show

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert n_\kappa - w_\kappa \Vert _{L^2(\Omega _T)}^2&= \varepsilon \iint _{\Omega _T} n_\kappa \partial _t w_\kappa - \iint _{\Omega _T} \big ( \tau n_\kappa \Delta w_\kappa + w_\kappa (\varepsilon \partial _t w_\kappa - \tau \Delta w_\kappa ) \big ) \\&= \varepsilon \int _{\Omega } ( n_\kappa (T) w_\kappa (T)- n_0 w_\kappa (0) ) - \varepsilon \iint _{\Omega _T} w_\kappa \partial _t n_\kappa \\&\quad - \iint _{\Omega _T} \big ( \tau n_\kappa \Delta w_\kappa + w_\kappa (\varepsilon \partial _t w_\kappa - \tau \Delta w_\kappa ) \big ) \\&= \varepsilon \int _{\Omega } ( n_\kappa (T) w_\kappa (T)- n_0 w_\kappa (0) ) + \varepsilon \iint _{\Omega _T} \big ( \nabla n_\kappa \cdot \nabla w_\kappa - n_\kappa \nabla c_\kappa \cdot \nabla w_\kappa \big ) \\&\quad + \tau \iint _{\Omega _T} \big ( \nabla n_\kappa \cdot \nabla w_\kappa - |\nabla w_\kappa |^2) - \frac{\varepsilon }{2} \int _\Omega (w_\kappa ^2(T)- w_\kappa ^2(0)), \end{aligned}$$\end{document}

where we have used the equation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} and integration by parts in the last computation. Recalling from Theorem 1.3 that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\kappa $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w_\kappa $$\end{document} are uniformly-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Omega _T)$$\end{document} that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \varepsilon \int _{\Omega } ( n_\kappa (T) w_\kappa (T)- n_0 w_\kappa (0) ) \right| \le \big (2 |\Omega | \Vert n_\kappa \Vert _{L^\infty (\Omega _T)} \Vert w_\kappa \Vert _{L^\infty (\Omega _T)} \big )\varepsilon \le C_T \varepsilon , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \frac{\varepsilon }{2} \int _\Omega (w_\kappa ^2(T)- w_\kappa ^2(0)) \right| \le \big ( |\Omega | \Vert w_\kappa \Vert _{L^\infty (\Omega _T)}^2 \big ) \varepsilon \le C_T \varepsilon . \end{aligned}$$\end{document}

Thanks to the uniform-in- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} boundedness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla n_\kappa $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla w_\kappa $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2((0,T);H^1(\Omega ))$$\end{document} , again from Theorem 1.3,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| \varepsilon \iint _{\Omega _T} \big ( \nabla n_\kappa \cdot \nabla w_\kappa - n_\kappa \nabla c_\kappa \cdot \nabla w_\kappa \big ) \right| \\&\quad \le \left( \Vert \nabla n_\kappa \Vert _{L^2(\Omega _T)} + \Vert n_\kappa \Vert _{L^\infty (\Omega _T)} \Vert \nabla c_\kappa \Vert _{L^2(\Omega _T)} \right) \Vert \nabla w_\kappa \Vert _{L^2(\Omega _T)} \varepsilon \\&\quad \le C_T \varepsilon , \end{aligned}$$\end{document}

as well as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \tau \iint _{\Omega _T} \big ( \nabla n_\kappa \cdot \nabla w_\kappa - |\nabla w_\kappa |^2) \right| \le \left( \Vert \nabla n_\kappa \Vert _{L^2(\Omega _T)} + \Vert \nabla w_\kappa \Vert _{L^2(\Omega _T)} \right) \Vert \nabla w_\kappa \Vert _{L^2(\Omega _T)} \tau \le C_T \tau . \end{aligned}$$\end{document}

Altogether, we get the estimate desired estimate.