The Keller-Segel system with logistic growth and signal-dependent motility

TL;DR

This paper analyzes a chemotaxis system with nonlinear motility functions, establishing global boundedness of solutions and exponential convergence to steady state under certain conditions.

Contribution

It proves the existence of globally bounded solutions and exponential convergence to steady state for a chemotaxis system with nonlinear motility functions.

Findings

Solutions are globally bounded for any initial data in W^{1,∞}.

Solutions exponentially converge to the steady state (1,1) if ur > K_0/16.

The analysis employs energy estimates and Lyapunov functions.

Abstract

The paper is concerned with the following chemotaxis system with nonlinear motility functions \begin{equation}\label{0-1}\tag{$\ast$} \begin{cases} u_t=\nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, 0=\Delta v+ u-v,& x\in \Omega, ~~t>0,\\ u(x,0)=u_0(x), & x\in \Omega, \end{cases} \end{equation} with homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset \R^2$ with smooth boundary, where the motility functions $\gamma(v)$ and $\chi(v)$ satisfy the following conditions \begin{itemize} \item {\color{black}$(\gamma,\chi)\in [C^2[0,\infty)]^2$} with $\gamma(v)>0$ and {\color{black} $\frac{|\chi(v)|^2}{\gamma(v)}$ is bounded for all $v\geq 0$.} %for all $v\geq 0$ and $\lim\limits_{v\to\infty}\frac{|\chi(v)|^2}{\gamma(v)}$ exists. \end{itemize} By employing the method of energy estimates , we establish the existence of globally bounded solutions of \eqref{0-1} with $\mu>0$ for any $u_0 \in W^{1, \infty}(\Omega)$. Then based on a Lyapunov function, we show that all solutions $(u,v)$ of \eqref{0-1} will exponentially converge to the unique constant steady state $(1,1)$ provided $\mu>\frac{K_0}{16}$ with $K_0=\max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)}$.