Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system

TL;DR

This paper establishes global existence, uniqueness, and regularity for a complex two-species reaction-diffusion system with nonlinear diffusion on volume and surface, utilizing entropic gradient structures to derive bounds.

Contribution

It introduces a novel approach to prove upper bounds for a reaction-diffusion system with nonlinear bulk and surface diffusion, ensuring well-posedness.

Findings

Proved global existence and uniqueness of solutions.

Established regularity results for the system.

Demonstrated the effectiveness of entropic gradient methods.

Abstract

We prove a global existence, uniqueness and regularity result for a two-species reaction-diffusion volume-surface system that includes nonlinear bulk diffusion and nonlinear (weak) cross diffusion on the active surface. A key feature is a proof of upper $L^{\infty}$-bounds that exploits the entropic gradient structure of the system.

Equations

$$\dot{u}-\mathrm{div}(\mu (u)\nabla u)$$

$$\mu (u)\frac{\partial u}{\partial \nu }$$

$$\mu (u)\frac{\partial u}{\partial \nu }+k\alpha (u^{\alpha }-\kappa v^{\beta })$$

$$\dot{v}-{\mathrm{div}}_{\Gamma }({\mu }_{\Gamma }(u,v){\nabla }_{\Gamma }v)-k\beta (u^{\alpha }-\kappa v^{\beta })$$

$${\mu }_{\Gamma }(u,v)\frac{\partial v}{\partial {\nu }_{\Gamma }}$$

$$\mu (x,u)=u^{\gamma }{\mathrm{Id}}_d,\mu (x,u)=e^{\delta u}{\mathrm{Id}}_d,\gamma ,\delta \in \mathbb{R}.$$

$${\mu }_{\Gamma }(u,v)=\frac{v}{\alpha u+\beta v}{\mathrm{Id}}_{d-1},$$

$${\mathrm{L}}^{q,q_{\Gamma }}$$

$${\mathrm{W}}^{1,q,q_{\Gamma }}$$

$${\mathrm{C}}^{\alpha ,{\alpha }_{\Gamma }}$$

$$W^{-1,q}(\Omega )={\left(W^{1,q^{\prime }}(\Omega )\right)}^{\prime },W^{-1,q_{\Gamma }}(\Gamma )={\left(W^{1,q_{\Gamma }^{\prime }}(\Gamma )\right)}^{\prime }$$

$${\mathrm{W}}^{-1,q,q_{\Gamma }}:={\left({\mathrm{W}}^{1,q^{\prime },q_{\Gamma }^{\prime }}\right)}^{\prime }.$$

$${\mathfrak{l}}_{u,v}(\psi ,\phi ):={\int }_{\Omega }\nabla {\psi }_u\cdot \mu (u)\nabla {\phi }_u\mathrm{d}x+{\int }_{\Gamma }{\nabla }_{\Gamma }{\psi }_v\cdot \mu (u,v)\nabla {\phi }_v\mathrm{d}y$$

$$\mathcal{F}(u,v)(\varphi)=\left(\begin{array}[]{l}\mathcal{F}_{u}(u,v)(\varphi)\\ \mathcal{F}_{v}(u,v)(\varphi)\end{array}\right)=\left(\begin{array}[]{l}-\alpha\int_{\Gamma}k(u^{\alpha}-\kappa v^{\beta})\varphi_{u}|_{\Gamma}\,\mathrm{d}y\\ +\beta\int_{\Gamma}k(u^{\alpha}-\kappa v^{\beta})\varphi_{v}\,\mathrm{d}y\end{array}\right),$$

$$\left(\begin{array}[]{c}\dot{u}\\ \dot{v}\end{array}\right)(t)+\mathcal{L}_{u(t),v(t)}^{q,q_{\Gamma}}(u(t),v(t))=\mathcal{F}(u(t),v(t))$$

$${\mathrm{MR}}_{q,q_{\Gamma }}^r:=L^r(J_T;{\mathrm{W}}^{1,q,q_{\Gamma }})\cap W^{1,r}(J_T;{\mathrm{W}}^{-1,q,q_{\Gamma }})$$

$${\mathrm{X}}_{q,q_{\Gamma }}^r:=({\mathrm{W}}^{1,q,q_{\Gamma }},{\mathrm{W}}^{-1,q,q_{\Gamma }}{)}_{1-\frac{1}{r},r}$$

$${\mathrm{MR}}_{q,q_{\Gamma }}^r\hookrightarrow C^0(J_T;{\mathrm{X}}_{q,q_{\Gamma }}^r).$$

$${\mathrm{X}}_{q,q_{\Gamma }}^r\hookrightarrow {\mathrm{C}}^{\beta ,{\beta }_{\Gamma }}\hookrightarrow {\mathrm{C}}^{0,0},$$

$$\mathrm{m}=\beta {\int }_{\Omega }u(t,x)\mathrm{d}x+\alpha {\int }_{\Gamma }v(t,y)\mathrm{d}y.$$

$$\mathrm{m}=\beta {\int }_{\Omega }{u}_0(x)\mathrm{d}x+\alpha {\int }_{\Gamma }{v}_0(y)\mathrm{d}y,$$

$$\mathrm{m}$$

$$0$$

$${\mathrm{MR}}_{q,q_{\Gamma }}^r\hookrightarrow C^{\delta }(J_T;{\mathrm{C}}^{\gamma ,{\gamma }_{\Gamma }})$$

$$\Vert (u_1,v_1){\Vert }_{{\mathrm{L}}^{\infty }},\Vert (u_2,v_2){\Vert }_{{\mathrm{L}}^{\infty }}\le L_{\infty },$$

$$\Vert \mathcal{F}(u_1,v_1)-\mathcal{F}(u_2,v_2){\Vert }_{{\mathrm{W}}^{-1,q,q_{\Gamma }}}$$

$$\le C(L_{\infty })\Vert (u_1-u_2),(v_1-v_2){\Vert }_{{\mathrm{L}}^{\infty }}$$

$$\le C(\tilde{L})\Vert (u_1-u_2),(v_1-v_2){\Vert }_{{\mathrm{X}}_{q,q_{\Gamma }}^r}.$$

$$0<c_u\le \frac{u_0}{u_{\ast }}\le C_u<+\infty ,0<c_v\le \frac{v_0}{v_{\ast }}\le C_v<+\infty$$

$${\left(\frac{u(t,x)}{u_{\ast }}\right)}^{\alpha },{\left(\frac{v(t,y)}{v_{\ast }}\right)}^{\beta }\le L,\text{ and }l\le u^{\alpha }(t,x),\kappa v^{\beta }(t,x),$$

$$f(u,v):$$

$$=\Lambda (u^{\alpha },\kappa v^{\beta })(\alpha \ln u-\beta \ln v-\ln \kappa )$$

$$=\Lambda (u^{\alpha },\kappa v^{\beta })(\alpha \ln \frac{u}{u_{\ast }}-\beta \ln \frac{v}{v_{\ast }}),$$

$$\Lambda (a,b)=\{\begin{array}{ll}\frac{a-b}{\ln a-\ln b},& \text{if }a,b>0,\text{a=b},\\ 0,& \text{if }a=0\text{ or }b=0,\\ a,& \text{if }a=b>0,\end{array}$$

$$f_0(u,v):=\{\begin{array}{ll}f(u,v)=\Lambda (u^{\alpha },\kappa v^{\beta })(\alpha \ln \frac{u}{u_{\ast }}-\beta \ln \frac{v}{v_{\ast }}),& \text{if }u,v>0,\\ 0,& \text{if }u\le 0\text{ or }v\le 0.\end{array}$$

$${\mu }_l^L(u):=\mu ({\hat{u}}_l^L),{\mu }_{\Gamma ,l}^l(u,v):={\mu }_{\Gamma }({\hat{u}}_l^L,{\hat{v}}_l^L),$$

Full text

Karoline Disser∗

FB Mathematik, TU Darmstadt,

Schlossgartenstr. 7,

64293 Darmstadt, Germany

1. Introduction

There has been a lot of recent interest in the mathematical analysis of reaction-diffusion processes that are coupled across volume and surface domains, due to their relevance in particular in the modelling of biological cells and technological processes, e.g. involving catalysis by surfactants [2, 6, 7, 9]. For systems coupled across volume and surface, a particular modelling issue and mathematical difficulty is the nonlinearity of diffusion that is typical of many biological processes (due to porous media, non-Newtonian flow, …) and naturally associated to diffusion that is modelled on multiple (spatial) scales ([1, 10, 12, 13, 14]). The aim of this paper is to prove a global well-posedness result for reaction-diffusion volume-surface systems that include typical general classes of nonlinear diffusion (like slow and fast diffusion) and nonlinear (weak) cross diffusion on the active surface. This generalizes some of the recent results on volume-surface reaction-diffusion [2, 6, 7] to the nonlinear diffusion case and to the case that only part of the boundary of the bulk domain is active.

Besides the well-posedness result based on the analysis for quasilinear volume-surface systems in [4], the aim of the paper is to introduce a proof of $L^{\infty}$-estimates that is different from the classical proof of comparison principles for the linearized part [6, 7]. Instead, we explicitly exploit the entropic gradient structure of the system [10] to show that the (chemical) logarithmic potentials remain pointwise bounded. This translates to pointwise estimates on the concentrations. A feature of this estimate is that it does not use the positivity of the (quasi)linear diffusion. The method was inspired by the boundedness-by-entropy method developed by Jüngel [11], but the argument here is more direct and thus does not promise as much generality. So it is unclear whether a strategy of this type can also be used to obtain results on more complex cases of multicomponent diffusion or reaction. This is the aim of future work.

1.1. Model equations

We consider the reaction-cross-diffusion system

[TABLE]

with initial conditions $u(0,x)=u_{0}(x)>0,x\in\Omega$ and $v(0,y)=v_{0}(y)>0,y\in\Gamma.$ Here, $\Omega\subset\mathbb{R}^{d}$,$d=2,3$ is a bounded strong Lipschitz domain, $\Gamma\subset\partial\Omega$ is an open connected part of its boundary, $\nu$ is the unit outer normal vector field at $\partial\Omega$ and if $d=3$, $\nu_{\Gamma}$ is the unit outer normal vector field at $\partial\Gamma$ (for a (canonical) definition and discussion of the operators $\mathrm{div}_{\Gamma}$ and $\nabla_{\Gamma}$ in this non-smooth case, see e.g. [5]). Locally uniformly in $(u,v)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$, the diffusion coefficients $\mu(\cdot,u)$, $\mu_{\Gamma}(\cdot,u,v)$ are measurable, bounded and elliptic. Uniformly in $x\in\Omega,y\in\Gamma$, $u\mapsto\mu(x,u)$ and $(u,v)\mapsto\mu_{\Gamma}(y,u,v)$ are locally Lipschitz. Examples are slow and fast diffusion like

[TABLE]

Correspondingly, these and expressions like

[TABLE]

are admissible for the diffusion coefficient on the surface. With natural boundary conditions, starting from positively bounded initial values, no degeneracy will develop for this system. The reversible chemical reaction on the surface is described by mass-action kinetics with forward and backward rates $k,\kappa>0$ and stoichiometric coefficients $\alpha,\beta\geq 1$.

1.2. Outline

Section 2 on preliminaries is mostly technical, in that it is used to introduce the notation and arguments needed to fit system (1) in the framework of the main result in [4] on existence and uniqueness for quasilinear dissipative bulk-interface dynamics. It comes before the main result as it is needed for the detailed/precise statement of the main result in Theorem 3.1 in Section 3. The $L^{\infty}$-estimate derived from the entropic gradient structure of system (1) is shown in Section 4.

2. Preliminary results and notation

Due to the quasilinear structure of system (1), global existence and uniqueness do not follow from standard linearization techniques, if only $L^{\infty}$ bounds are available globally a priori. To see that system (1) fits with the theory in [4], for convenience and in order to give a precise statement of the main result, we introduce some of the functional analytic framework used for volume-surface systems in [4].

2.1. Notation and function spaces

For $q,q_{\Gamma}\in[1,\infty]$ and $0\leq\alpha,\alpha_{\Gamma}<1$, define

[TABLE]

with components $\mathrm{L}^{q,q_{\Gamma}}\ni\varphi=(\varphi_{u},\varphi_{v})$, where $L^{q}$, $W^{1,q}$, $C^{\alpha}$ denote the usual Lebesgue, Sobolev and Hölder spaces. In the following, $q^{\prime}$ is the dual exponent of $q$ with $1=\frac{1}{q}+\frac{1}{q^{\prime}}$ and we denote dual Sobolev spaces by

[TABLE]

and

[TABLE]

Let $(u,v)\in\mathrm{C}^{0,0}$ with $u,v>0$. Then using the assumptions on $\mu,\mu_{\Gamma}$, the bilinear form $\mathfrak{l}_{u,v}\colon\mathrm{W}^{1,2,2}\times\mathrm{W}^{1,2,2}\to\mathbb{R}$ given by

[TABLE]

is well-defined, continuous and coercive (after shifting). It induces the two-component Neumann Laplacian $\mathcal{L}_{u,v}\colon\mathrm{W}^{1,2,2}\to\mathrm{W}^{-1,2,2}$ with $\mathcal{L}_{u,v}(\psi)(\varphi):=\mathfrak{l}_{u,v}(\psi,\varphi)$. For $q,q_{\Gamma}\in[2,\infty)$, let $\mathcal{L}_{u,v}^{q,q_{\Gamma}}$ be the closed and densely defined restriction of $\mathcal{L}_{u,v}$ to $\mathrm{W}^{-1,q,q_{\Gamma}}$.

2.2. Interpretation of (1)

In order to re-write system (1) as a quasilinear abstract Cauchy problem in $\mathrm{W}^{-1,q,q_{\Gamma}}$, consider the reaction terms as components of a functional $\mathcal{F}(u,v)\in\mathrm{W}^{-1,q,q_{\Gamma}}$ through

[TABLE]

for all $\varphi\in\mathrm{W}^{1,q^{\prime},q_{\Gamma}^{\prime}}$. With the existence of the traces $u|_{\Gamma}\in C^{0}(\Gamma)$ and $\varphi_{u}|_{\Gamma}\in L^{q^{\prime}}(\Gamma)$, $\mathcal{F}(u,v)$ is well-defined for all $t>0$. Note that this is just the usual way of casting inhomogeneous Neumann boundary conditions in a weak form. Hence, system (1) can be written as the abstract Cauchy problem

[TABLE]

in $\mathrm{W}^{-1,q,q_{\Gamma}}$ with initial data $(u,v)(0)=(u_{0},v_{0})$. Due to the uniqueness and regularity of solutions (Theorem 3.1), solutions of (2) solve system (1) in an adequate sense.

2.3. Regularity results

To solve (2) on any time interval $J_{T}=(0,T),T>0$, we use that for $r\in(1,\infty)$, $\mathcal{L}_{u,v}^{q,q_{\Gamma}}$ has maximal parabolic regularity in $L^{r}(J_{T};\mathrm{W}^{-1,q,q_{\Gamma}})$ and that there are $q>d$, $q_{\Gamma}>2$ such that $\mathrm{dom}(\mathcal{L}_{u,v}^{q,q_{\Gamma}}+\lambda)=\mathrm{W}^{1,q,q_{\Gamma}}$ for $\lambda>0$ (for details and proofs, see [4][Sect. 2.3]).

For this, if $d=3$, we use the following additional assumption on $\mu$: it is of the form $\mu(x,u)=\mu_{\circ}(u)m(x)$, where $\mu_{\circ}\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a scalar function that is locally Lipschitz and $m\colon\Omega\to\mathbb{R}_{>0,\mathrm{sym}}^{3\times 3}$ is continuous (for a slight generalization, see [4][Sect. 2.2(4)]).

We define

[TABLE]

as the space of solutions corresponding to the setting of system (2) and

[TABLE]

as the corresponding time trace space, satisfying

[TABLE]

The following embedding result is essential [4][Lemma 2.4(2)]: If $q>d$, $q_{\Gamma}>d-1$ and $r>\max(\frac{2q}{q-d},\frac{2q_{\Gamma}}{q_{\Gamma}-d+1})$, then

[TABLE]

for $0<\beta\leq 1-\frac{d}{q}-\frac{2}{r}$ and $0<\beta_{\Gamma}\leq 1-\frac{d-1}{q_{\Gamma}}-\frac{2}{r}$. In particular, this implies that for any $(u,v)\in\mathrm{MR}_{q,q_{\Gamma}}^{r}$, at any $t>0$, the linearized operator $\mathcal{L}_{u(t),v(t)}^{q,q_{\Gamma}}$ is well-defined.

2.4. Equilibria

Note that the system (1) preserves the total mass

[TABLE]

Given initial values $(u_{0},v_{0})\geq 0$ with mass

[TABLE]

the Wegscheider conditions

[TABLE]

determine uniquely the equilibrium $(u_{*},v_{*})\in\mathbb{R}_{+}^{2}$ corresponding to $(u_{0},v_{0})$ (to see this here, note that the function $v\mapsto\beta|\Omega|\kappa^{1/\alpha}v^{\beta/\alpha}+\alpha|\Gamma|v_{*}$ is strictly monotone and thus invertible from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$.)

3. Main result

Theorem 3.1**.**

There exist $q>d$, $q_{\Gamma}>2$ such that for all $r>\max\left(\frac{2q}{q-d},\frac{2q_{\Gamma}}{q_{\Gamma}-d+1}\right)$, and for all positive $(u_{0},v_{0})\in\mathrm{X}_{q,q_{\Gamma}}^{r}$, there exists a unique global positive solution $(u,v)\in\mathrm{MR}_{q,q_{\Gamma}}^{r}$ of system (2).

The proof is based on the main result in [4] that provides global existence and uniquenss in the functional analytic framework of system (2) if the right-hand-side $\mathcal{F}$ is Lipschitz and preserves $L^{\infty}$-bounds (a posteriori). Regarding regularity of solutions, note that

[TABLE]

for suitable $\delta,\gamma,\gamma_{\Gamma}>0$ [4][Lemma 2.4(3)] and note that $\mathrm{W}^{1,q,q_{\Gamma}}\hookrightarrow\mathrm{X}_{q,q_{\Gamma}}^{r}$ by definition (as an upper limit on the regularity required of the initial data – with parabolic smoothing, less than $\mathrm{X}_{q,q_{\Gamma}}^{r}$ can be shown to be sufficient).

Proof.

To be precise, we first note how the assumptions of Theorem 3.1 in [4] can be satisfied (there may be several choices): in [4], set $m_{\Gamma}=m_{+}=m_{-}=0$, $\Omega_{+}=\Omega$, $\Omega_{-}=\emptyset$, $u_{+}=u$, $u_{\Gamma}=v$, $f_{+}=h_{+}=h_{\Gamma}=0$, $g_{+}(u,v)=-k\alpha(u^{\alpha}-\kappa v^{\beta})$, $f_{\Gamma}(u,v)=k\beta(u^{\alpha}-\kappa v^{\beta})$, $k_{+}(u)=\mu(u)$, and $k_{\Gamma}(u,v)=\mu_{\Gamma}(u,v)$. Then the assumptions in Section 2.1 in [4] are satisfied and it remains to check Assumption 2.5 there (local Lipschitzianity of $\mathcal{F}$): for $(u_{1},v_{1}),(u_{2},v_{2})\in\mathrm{X}_{q,q^{\Gamma}}^{r}$ with $\|(u_{1},v_{1})\|_{\mathrm{x}_{q,q_{\Gamma}}^{r}},\|(u_{2},v_{2})\|_{\mathrm{x}_{q,q_{\Gamma}}^{r}}\leq\tilde{L}$, we have

[TABLE]

by embedding (3), so

[TABLE]

The global result Theorem 3.1 below follows from Theorem 3.1 in [4] as in Corollary 3.6 there, using Lemma 4.1 below on $L^{\infty}$-bounds. In fact, a Schaefer fixed point argument is used for the proof of existence of solutions. It requires uniform $L^{\infty}$ -bounds for solutions of system (2) with right-hand-side $\lambda\mathcal{F}(u,v)$, $0<\lambda\leq 1$ and initial value $\lambda(u_{0},v_{0})$. Clearly, this also follows from the proof of Lemma (4.1). ∎

4. $L^{\infty}$-estimates based on the entropic gradient structure

The main result of this section is the following Lemma. It is sufficient for proving Theorem 3.1, but also the aim is to introduce a method that derives pointwise upper bounds from the entropy-producing structure of the reaction, without using the diffusivity.

Lemma 4.1**.**

Let $r,q,q_{\Gamma}$ be as in Theorem 3.1 and let $0<(u_{0},v_{0})\in\mathrm{X}_{q,q_{\Gamma}}^{r}$. Then the corresponding equilibrium $(u_{*},v_{*})$ is positive and there are constants $c_{u},c_{v},C_{u},C_{v}>0$ such that

[TABLE]

for all $x\in\Omega,y\in\Gamma$. Assume that $(u,v)\in\mathrm{MR}_{q,q_{\Gamma}}^{r}$ is a solution of (2) with initial data $(u_{0},v_{0})$. Define $l:=\min(c_{u}^{\alpha},\kappa c_{v}^{\beta})$ and $L:=\max(C_{u}^{\alpha},C_{v}^{\beta})$, then

[TABLE]

for all $t\geq 0$, $x\in\Omega$ and $y\in\Gamma$.

Proof.

First note that since $\mathrm{m}>0$, $(u_{*},v_{*})$ is positive, and the embedding (3) implies that $(u_{0},v_{0})$ are positively bounded from above and below.

To show that the reaction in (1) is entropy-producing, the reaction rate is reformulated as

[TABLE]

where the last equality follows from (4). Here,

[TABLE]

denotes the logarithmic mean. The equality in (7) is justified if $u,v\geq 0$. Since $u,v\geq 0$ will hold a posteriori (even a uniform positive lower bound), the rate function $f(u,v)=k(u^{\alpha}-\kappa v^{\beta})$ can be replaced with

[TABLE]

We do not know positivity of solutions a priori, so to be precise, we also use the following very small trick: given $0<(u_{0},v_{0})\in\mathrm{C}^{0,0}$ and the corresponding equilibrium $(u_{*},v_{*})>0$, we show the bounds (6) a posteriori, hence, a priori, in (2), the diffusion coefficients $\mu,\mu_{\Gamma}$ can be replaced by

[TABLE]

where

[TABLE]

and

[TABLE]

By the assumptions in Subsection 1.1, $\mu_{l}^{L}$ and $\mu_{\Gamma,l}^{L}$ are uniformly bounded and elliptic with

[TABLE]

for a.e. $x\in\Omega$ and $y\in\Gamma$, for all $X,X^{\prime}\in\mathbb{R}^{d}$ and $Y,Y^{\prime}\in\mathbb{R}^{d-1}$, and for all $(u,v)\in\mathbb{R}^{2}$ .

Now we actually show that the adapted system (2) with $\mu_{l}^{L},\mu_{\Gamma,l}^{L}$ has a unique global regular solution $(u,v)$ that satisfies (6). But then,

[TABLE]

and

[TABLE]

for all $t\geq 0$. Due to the Hölder regularity of $(u,v)$ in space and time (5) and due to the local Lipschitzianity of $f$, $\mu$, and $\mu_{\Gamma}$, $(u,v)$ is also the unique solution of the original system (2).

The upper bounds in Lemma 4.1 are shown using the entropy-producing structure of the system. In the usual proof of a comparison principle, the dissipation of the $L^{2}$-energy of truncated solutions is used to show pointwise bounds. Here, the idea is that, similarly, the dissipation of the free energy of truncated potentials provides pointwise bounds.

For this method to work, the solution must be constructed to have an entropy estimate and it must be sufficiently regular to allow for the truncation. Here, we are dealing with (fairly) regular solutions and it is straightforward to do everything rigorously (see (LABEL:eq:entropyProduction) below). Regarding weaker notions of solutions, in [8], the corresponding relative entropy estimate is shown for a general class of reaction-diffusion systems, for renormalized (and thus for weak) solutions. A similar result should hold for this simple volume-surface case, even with nonlinear diffusion (the truncated version may not follow automatically for weak solutions though). In [6], the entropy estimate is assumed and used to show the exponential equilibration of the linear diffusion volume-surface system.

Let $\mathcal{E}$ be the (negative) relative entropy for the system given by

[TABLE]

where

[TABLE]

Moreover, let

[TABLE]

a relative entropy adapted to the bound $L$. Then define

[TABLE]

so that

[TABLE]

and

[TABLE]

With the definition

[TABLE]

and

[TABLE]

this can be written as

[TABLE]

Due to $(u,v)\in\mathrm{MR}_{q,q_{\Gamma}}^{r}$ and since $e^{\prime}$ is uniformly Lipschitz on $(u_{*},+\infty)$, $(v_{*},+\infty)$,

[TABLE]

so apply equation (2) to $(\xi^{L},\chi^{L})$ and add both components to get that, for a.e. $t>0$,

[TABLE]

where $\Gamma_{0}\subset\Gamma$ is the subset of $\Gamma$, where $(u,v)\geq 0$ . If

[TABLE]

for all $t>0$, then

[TABLE]

hence,

[TABLE]

for all $t>0$ ($\mathcal{E}_{L}$ is a strict Lyapunov functional), and then the upper bound in Lemma 4.1 is proved. To show this, the terms on the right-hand-side of (LABEL:eq:entropyProduction) can be considered in a pointwise manner. First the reaction term: for each $t>0$, let

[TABLE]

then $\ln\left(\frac{u}{u_{*}}\right)^{\alpha}-\ln\left(\frac{v}{v_{*}}\right)^{\beta}>0$ in $\Gamma_{\xi}$ and $\ln\left(\frac{u}{u_{*}}\right)^{\alpha}-\ln\left(\frac{v}{v_{*}}\right)^{\beta}<0$ in $\Gamma_{\chi}$, so

[TABLE]

where in the last line, the adaptation of $\mathcal{E}$ with the weights $L^{1/\alpha}$, $L^{1/\beta}$ was used. This would not be necessary if the case that both $u$ and $v$ are pointwise large can be excluded a priori, an estimate that is often known and employed in the linear diffusion case, see [15]. In summary,

[TABLE]

for the reaction term in (LABEL:eq:entropyProduction).

For the diffusion, the estimate is

[TABLE]

and, in the same way,

[TABLE]

This proves the upper bound. The uniform lower bound can be proved by a standard comparison principle (dissipation for the $L^{2}$-norm): Let $\sigma_{u}:=l^{1/\alpha}$ and $\sigma_{v}:=\kappa l^{1/\beta}$, then $\sigma_{u}\leq u_{0}$ and $\sigma_{v}\leq v_{0}$ and $(\sigma_{u},\sigma_{v})$ is a (stationary) solution of system (1). Define

[TABLE]

then $(u_{-},v_{-})(0)=0$ and $(u_{-},v_{-})\in L^{r^{\prime}}(J_{T};\mathrm{W}^{1,q^{\prime},q^{\prime}_{\Gamma}})$ is a suitable test function for (2), so that

[TABLE]

We estimate the reaction term in detail: for each $t>0$, let

[TABLE]

then for some (generic) constant $C>0$,

[TABLE]

where $(*)$ follows from the mean value theorem and $\sigma_{u}>u$, and the last inequality follows from Young’s inequality and the trace estimate $\int_{\Gamma}|u_{-}|^{2}\,\mathrm{d}y\leq C\|u_{-}\|_{H^{1}(\Omega)}.$ Similarly,

[TABLE]

Hence,

[TABLE]

and thus, by Gronwall’s inequality, $(\sigma_{u},\sigma_{v})\leq(u,v)(t)$ for all $t\geq 0$. ∎