Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system

TL;DR

This paper extends previous results on lower bounds for blow-up time in a chemotaxis system by considering nonlinear diffusion and non-convex domains, using sharp inequalities for improved estimates.

Contribution

It generalizes earlier work to include nonlinear diffusion effects and non-convex domains, providing a more comprehensive lower bound for blow-up time.

Findings

Derived a new lower bound for blow-up time in the chemotaxis system.

Showed the effect of nonlinear diffusion on blow-up behavior.

Removed the convexity restriction on the domain in the analysis.

Abstract

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system \begin{equation*} \begin{cases} u_t=\nabla \cdot [(u+\alpha)^{m_1-1} \nabla u-\chi u(u+\alpha)^{m_2-2} \nabla v] & {\rm in} \; \Omega \times (0,T), \\[1mm] v_t=\Delta v-v+u & {\rm in} \; \Omega \times (0,T) \end{cases} \end{equation*} under Neumann boundary conditions and initial conditions, where $\Omega$ is a general bounded domain in $\mathbb{R}^n$ with smooth boundary, $\alpha>0$, $\chi>0$, $m_1, m_2 \in \mathbb{R}$ and $T>0$. Recently, Anderson-Deng (2017) gave a lower bound for the blow-up time in the case that $m_1=1$ and $\Omega$ is a convex bounded domain. The purpose of this paper is to generalize the result in Anderson-Deng (2017) to the case that $m_1 \neq 1$ and $\Omega$ is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo-Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of $\Omega$.

Tables

Table 1. Table 1: The known results on lower bounds for the blow-up time in ( 1.1 )

No work		No work	No work

Equations

$$\{\begin{array}{ll}{u}_t=\nabla \cdot [(u+\alpha {)}^{m_1-1}\nabla u-\chi u(u+\alpha {)}^{m_2-2}\nabla v]& \mathrm{in}\Omega \times (0,T),\\ v_t=\Delta v-v+u& \mathrm{in}\Omega \times (0,T)\end{array}$$

$$\{\begin{array}{ll}{u}_t=\nabla \cdot [(u+\alpha {)}^{m_1-1}\nabla u-\chi u(u+\alpha {)}^{m_2-2}\nabla v]& \mathrm{in}\Omega \times (0,T),\\ v_t=\Delta v-v+u& \mathrm{in}\Omega \times (0,T),\\ \nabla u\cdot \nu =\nabla v\cdot \nu =0& \mathrm{on}\partial \Omega \times (0,T),\\ u(\cdot ,0)=u_0,v(\cdot ,0)=v_0& \mathrm{in}\Omega ,\end{array}$$

$$\alpha >0,\chi >0,m_1,m_2\in \mathbb{R}.$$

$${\int }_{\Omega }u(\cdot ,t)={\int }_{\Omega }{u}_0$$

$${\tilde{t}}^{\ast }\ge {\int }_{{\Phi }_1(0)}^{\infty }\frac{d\xi }{V{\xi }^{\frac{3}{2}}+W{\xi }^2}(n=2)$$

$${\tilde{t}}^{\ast }\ge {\int }_{{\Phi }_1(0)}^{\infty }\frac{d\xi }{X{\xi }^{\frac{3}{2}}+Y{\xi }^3}(n=3);$$

$${\Phi }_1(t):=\kappa {\int }_{\Omega }{u(\cdot ,t)}^2+{\int }_{\Omega }|\Delta v(\cdot ,t){|}^2(t>0)$$

$${\Phi }_2(t):={\int }_{\Omega }{\left(u(\cdot ,t)+1\right)}^p+{\int }_{\Omega }{\left|\nabla v(\cdot ,t)\right|}^{2q}(t>0)$$

$$\{\begin{array}{ll}{u}_t=\nabla \cdot [\nabla u-k_1(t)u^{m_2-1}\nabla v]+\text{boldmath}f(u)& \mathrm{in}\Omega \times (0,T),\\ v_t=k_2(t)\Delta v-k_3(t)v+k_4(t)u& \mathrm{in}\Omega \times (0,T)\end{array}$$

$$m_1$$

$$\Vert u(\cdot ,t){\Vert }_{L^p(\Omega )}\le |\Omega {|}^{\frac{1}{p}}\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}.$$

$$u\in C\left(\overline{\Omega }\times [0,T)\right)\cap C^{2,1}\left(\overline{\Omega }\times (0,T)\right),$$

$$v\in C\left(\overline{\Omega }\times [0,T)\right)\cap C^{2,1}\left(\overline{\Omega }\times (0,T)\right)\cap L_{\mathrm{loc}}^{\infty }\left([0,T);W^{1,\infty }(\Omega )\right)$$

$$\underset{t\nearrow t^{\ast }}{\lim }(\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert v(\cdot ,t){\Vert }_{W^{1,\infty }(\Omega )})=\infty .$$

$$p>\max \left\{\frac{n}{2}(m_2-m_1),n(m_2-m_1-1),n\right\},$$

$$p>\max \left\{\frac{q(2m_2-m_1-3)}{q\eta -q-1},-2m_2+m_1+3,\frac{2q}{q\eta -q+1},\frac{\eta (m_1-1)}{(\eta -1)(\eta -2)}\right\},$$

$$\{\begin{array}{ll}\eta \in (1,2)\text{is any}& (n=1,2),\\ \eta :=\frac{n}{n-1}& (n\ge 3).\end{array}$$

$$t^{\ast }\ge {\int }_{\Phi (0)}^{\infty }\frac{d\tau }{A{\tau }^{f(\eta ,r)}+B{\tau }^{f(\eta ,1)}+C{\tau }^{\eta }+D},$$

$$\Phi (t):=\frac{1}{p}{\int }_{\Omega }{\left(u(\cdot ,t)+\alpha \right)}^p+\frac{1}{q}{\int }_{\Omega }|\nabla v(\cdot ,t){|}^{2q}(t>0),$$

$$f(\eta ,s):=1+\frac{\eta -1}{n\left(\frac{1}{n}-\frac{\eta }{2}+\frac{1}{2s}\right)}(s>0),$$

$$r=r(m_1):=\frac{p}{p+m_1-1},$$

$$s\ge 1⟹t^{\ast }\ge \frac{1}{f(\eta ,r)-1}\cdot \frac{\Phi (0)}{A{\Phi (0)}^{f(\eta ,r)-1}+B{\Phi (0)}^{f(\eta ,1)-1}+C{\Phi (0)}^{\eta -1}+D},$$

$$s<1⟹t^{\ast }\ge \frac{1}{f(\eta ,1)-1}\cdot \frac{\Phi (0)}{B{\Phi (0)}^{f(\eta ,1)-1}+A{\Phi (0)}^{f(\eta ,r)-1}+C{\Phi (0)}^{\eta -1}+D}.$$

$$\frac{d\Phi }{dt}+\frac{p-1}{2}{\left(\frac{2}{p+m_1-1}\right)}^2{\int }_{\Omega }{\left|\nabla (u+\alpha {)}^{\frac{p+m_1-1}{2}}\right|}^2+\left(\frac{2(q-1)}{q^2}-\delta \right){\int }_{\Omega }{\left|\nabla |\nabla v{|}^q\right|}^2$$

$$\le \frac{{\chi }^2(p-1)}{2}{\int }_{\Omega }(u+\alpha {)}^{p+2m_2-m_1-3}|\nabla v{|}^2+\frac{4(q-1)+n}{2}{\int }_{\Omega }(u+\alpha {)}^2|\nabla v{|}^{2q-2}+D_{\delta }$$

$${\int }_{\Omega }{\left(u+\alpha \right)}^{p\eta }\le C{\left({\int }_{\Omega }(u+\alpha {)}^p+{\int }_{\Omega }|\nabla (u+\alpha {)}^p|\right)}^{\eta }$$

$${\int }_{\Omega }{\left(u+\alpha \right)}^{p\eta }\le c{\Vert \nabla (u+\alpha {)}^{\frac{p}{2}}\Vert }_{L^2(\Omega )}^{\eta }{\Vert (u+\alpha {)}^{\frac{p}{2}}\Vert }_{L^2(\Omega )}^{\eta }+\tilde{c}{\Vert (u+\alpha {)}^{\frac{p}{2}}\Vert }_{L^2(\Omega )}^{2\eta }$$

$$\Vert w{\Vert }_{L^p(\Omega )}\le c\left(\Vert \nabla w{\Vert }_{L^r(\Omega )}^a\Vert w{\Vert }_{L^q(\Omega )}^{1-a}+\Vert w{\Vert }_{L^s(\Omega )}\right)$$

$${\int }_{\partial \Omega }|\nabla w{|}^{2q-2}\frac{\partial |\nabla w{|}^2}{\partial \nu }\le \delta {\int }_{\Omega }{\left|\nabla |\nabla w{|}^q\right|}^2+C_{\delta }.$$

$$\frac{\partial |\nabla w{|}^2}{\partial \nu }\le 0\mathrm{on}\partial \Omega .$$

$$T_{\max }=\infty \text{or}\underset{t\nearrow T_{\max }}{\mathrm{limsup}}\left(\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert v(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\right)=\infty .$$

$$\Vert u(\cdot ,t){\Vert }_{L^p(\Omega )}\le C\text{for all}t\in (0,T).$$

Full text

0002010Mathematics Subject Classification. Primary: 35B44, 35K51, 35K59; Secondary: 35Q92, 92C17. 000Key words and phrases: blow-up time; chemotaxis system; nonlinear diffusion.

**Effect of nonlinear diffusion on a lower bound

for the blow-up time

in a fully parabolic chemotaxis system **

Teruto Nishino000E-mail addresses: teruto.nishino@gmail.com, yokota@rs.kagu.tus.ac.jp

Tomomi Yokota***Corresponding author. †††Partially supported by Grant-in-Aid for Scientific Research (C), No. 25400119.

Department of Mathematics

Tokyo University of Science

• Abstract. This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system

[TABLE]

under Neumann boundary conditions and initial conditions, where $\Omega$ is a general bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $\alpha>0$, $\chi>0$, $m_{1},m_{2}\in\mathbb{R}$ and $T>0$. Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that $m_{1}=1$ and $\Omega$ is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that $m_{1}\neq 1$ and $\Omega$ is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of $\Omega$.

1 Introduction

In this paper we consider a lower bound for the blow-up time in the following fully parabolic chemotaxis system with nonlinear diffusion:

[TABLE]

where $\Omega$ is a general bounded domain in $\mathbb{R}^{n}$ ($n\in\mathbb{N}$) with smooth boundary $\partial\Omega$ and $\nu$ is the outward normal vector to $\partial\Omega$ and $T>0$. The initial data $u_{0}$ and $v_{0}$ are supposed to be nonnegative functions such that $u_{0}\in C(\overline{\Omega})$ and $v_{0}\in C^{1}(\overline{\Omega})$. Also we assume that

[TABLE]

In the system (1.1), the unknown function $u(x,t)$ represents the density of the cell population and the unknown function $v(x,t)$ shows the concentration of the signal substance at place $x$ and time $t$. The system (1.1) with the simplest choices $m_{1}=1$ and $m_{2}=2$ describes a part of life cycle of cellular slime molds with chemotaxis and it was proposed by Keller–Segel [18] in 1970. After that, a quasilinear system such as (1.1) was proposed by Painter–Hillen [25]. A number of variations of the original Keller–Segel system are proposed and investigated (see e.g., Bellomo–Bellouquid–Tao–Winkler [2], Hillen–Painter [9] and Horstmann [10, 11]).

According to a continuity model, the first equation in (1.1) has the flux vector $F=-\left[(u+\alpha)^{m_{1}-1}\nabla u-\chi u(u+\alpha)^{m_{2}-2}\nabla v\right]$. We can recognize that $(u+\alpha)^{m_{1}-1}\nabla u$ represents the diffusive flux and $-\chi u(u+\alpha)^{m_{2}-2}\nabla v$ represents the chemotactic flux modelling undirected cell migration and the advective flux with velocity dependent on the gradient of the signal. More precisely, when cellular slime molds plunge into hunger, they move towards higher concentrations of the chemical substance secreted by cells.

From a mathematical point of view, $u$ in (1.1) enjoys the mass conservation property:

[TABLE]

for all $t\in(0,T)$. It is a meaningful question whether solutions of (1.1) remain bounded or blow up. As to this question, it is known that the borderline between boundedness and blow-up is the case that $m_{2}=m_{1}+\frac{2}{n}$, $m_{1}\geq 1$, $m_{2}\geq 2$. According to the result established by Horstmann–Winkler [12, Theorems 4.1 and 6.1] in the case $m_{1}=1$, it can be expected that (1.1) has a global bounded solution in the case that $m_{2}<m_{1}+\frac{2}{n}$ and a blow-up solution in the case that $m_{2}>m_{1}+\frac{2}{n}$. Indeed, in the case that $\Omega$ is a bounded domain and $m_{2}<m_{1}+\frac{2}{n}$, there exists a global bounded solution of (1.1) (see Tao–Winkler [33], Ishida–Seki–Yokota [13] and Senba–Suzuki [31]). In addition, this result was shown also for the degenerate chemotaxis system ((1.1) with $\alpha=0$) (see Ishida–Yokota [14, 16] when $\Omega=\mathbb{R}^{n}$ and $m_{2}<m_{1}+\frac{2}{n}$; [13] when $\Omega$ is a bounded domain and $m_{2}<m_{1}+\frac{2}{n}$; Mimura [23] when $\Omega$ is a bounded domain with Dirichlet–Neumann boundary condition, $m_{2}=2$ and $m_{1}>2-\frac{2}{n}$). If $m_{2}\geq m_{1}+\frac{2}{n}$, then the results are divided by the size of initial data. For example, the system (1.1) has a global solution with small initial data when $\Omega=\mathbb{R}^{n}$ and $\alpha=0$ even if $m_{2}\geq m_{1}+\frac{2}{n}$ (see Ishida–Yokota [15]). On the other hand, in the case that \Omega=B_{R}:=\{x\in\mathbb{R}^{n}\big{|}|x|<R\} $(R>0)$, $m_{1}=1$, $m_{2}=2$, $n\geq 3$, which implies $m_{2}>m_{1}+\frac{2}{n}$, there exist initial data such that the radially symmetric solution of (1.1) blows up in finite time (see Winkler [34]). The result was extended to the case that $\Omega=B_{R}$, $m_{2}>m_{1}+\frac{2}{n}$, $n\geq 2$ (see Cieślak–Stinner [3, 4] when $\alpha>0$, Hashira–Ishida–Yokota [8] when $\alpha=0$). In the most important case that $\Omega=B_{R}$, $m_{1}=1$, $m_{2}=2$, $n=2$, which implies $m_{2}=m_{1}+\frac{2}{n}$, there exist initial data such that the corresponding solutions of (1.1) blow up in finite time (see Mizoguchi–Winkler [24]).

We are especially interested in a *lower bound *for the blow-up time for solutions of (1.1), because it seems to be important to know how $m_{1}$ affects on the blow-up time for solutions of (1.1). The study of a lower bound for the blow-up time seems to be interesting widely for general parabolic systems (see Payne–Schaefer [27] and Enache [5]), wave equations (see Philippin [30]) and heat equations (see Payne–Philippin–Vernier Piro [26]). Moreover, explicit lower bounds for the blow-up time for solutions of various semilinear parabolic equations were obtained by [27]. As to chemotaxis systems, Payne–Song [28, 29] established a lower bound of blow-up time for solutions of (1.1) with $m_{1}=1$ and $m_{2}=2$ in the form

[TABLE]

and

[TABLE]

note that $\widetilde{t}^{\ast}$ means the blow-up time in $\Phi_{1}$-measure, i.e., $\lim_{t\nearrow\widetilde{t}^{\ast}}\Phi_{1}(t)=\infty$, where $\Phi_{1}(t)$ is defined as

[TABLE]

with some $\kappa>0$. When $\Omega$ is a convex bounded domain and $m_{1}=1$, Li–Zheng [19] gave a lower bound for the blow-up time for solutions of (1.1) by using (1.2) in the case that $m_{2}\in(\frac{5}{3},2]$, $n=3$ and in the case that $m_{2}\in[2,3)$, $n=2$. After that, when $\Omega=B_{1}$, $\alpha=1$ and $m_{1}=1$, in the case that $m_{2}\in[\frac{5}{3},3]$ and $n=3$, Tao–Vernier Piro [32] introduced the measure $\Phi_{2}(t)$ in the form

[TABLE]

for suitable $p,q>1$ ($p=2$ and $q=4$ when $m_{2}\in[\frac{5}{3},2]$; $p=5$ and $q=22$ when $m_{2}\in(2,3]$) from the view point of local existence of classical solutions to (1.1) and an initial datum $v_{0}\in W^{1,q}(\Omega)$ (see [2, Lemma 3.1]). This restriction on $m_{2}$ and $n$ was removed by Anderson–Deng [1] when $\Omega$ is a convex bounded domain and $m_{1}=1$. Furthermore, as a new attempt to estimating a lower bound for the blow-up time in the above sense, Marras–Vernier Piro–Viglialoro [21, 22] obtained a lower bound for the blow-up time of the more generalized equation with a source term:

[TABLE]

under Neumann boundary conditions and initial conditions, where $k_{i}(t)\ (i=1,2,3,4)$ are nonnegative smooth functions of $t$, $m_{2}\in[2,3)$ when $n=2$, $m_{2}\in(\frac{5}{3},2)$ when $n=3$, $f$ satisfies $f(u)\leq cu^{2}$ with $c>0$. A similar result for the parabolic–elliptic version of (1.4) was deduced by Jiao–Zeng [17].

Now we focus on the studies obtained by [32] and [1] which gave a lower bound for the blow-up time for solutions of (1.1) under the following conditions:

• •

“$m_{1}=1$”, $m_{2}\in[\frac{5}{3},3]$, $n=3$, $\Omega$ is a unit ball $B_{1}\subset\boldmath\mathbb{R}^{3}$ ([32]);

• •

“$m_{1}=1$”, $m_{2}\in\mathbb{R}$, $n\in\mathbb{N}$, $\Omega$ is a *convex *bounded domain in $\mathbb{R}^{n}$ ([1]).

However, there is still room for improvements in these results. More precisely, we cannot find any results in the nonlinear case that $m_{1}\neq 1$ and $\Omega$ is a non-convex bounded domain. Hence the current situation is summarized in Table 1.