arXiv 2005.114622020-05-23math.AP

具有逻辑增长和信号依赖运动的Keller-Segel系统

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

Hai-Yang Jin, Zhi-An Wang

摘要：

本文研究了具有光滑边界的有界区域$Ω\subset \R^2$中具有齐次Neumann边界条件的具有非线性运动函数\begin{equation}\label{0-1}\tag{$\ast$} \begin{cases} u_t=\nabla \cdot (γ(v)\nabla u- uχ(v)\nabla v)+μu(1-u), &x\in Ω, ~~t>0, 0=Δv+ u-v,& x\in Ω, ~~t>0,\\ u(x,0)=u_0(x), & x\in Ω, \end{cases} \end{equation}的趋化系统，其中运动函数$γ(v)$和$χ(v)$满足以下条件：\begin{itemize} \item {\color{black}$(γ,χ)\in [C^2[0,\infty)]^2$} with $γ(v)>0$ and {\color{black} $\frac{|χ(v)|^2}{γ(v)}$ is bounded for all $v\geq 0$.} %for all $v\geq 0$ and $\lim\limits_{v\to\infty}\frac{|χ(v)|^2}{γ(v)}$ exists. \end{itemize}.利用能量估计的方法，对任意$u_0 \in W^{1, \infty}(Ω)$，建立了具有$μ>0$的\eqref{0-1}的整体有界解的存在性.然后基于一个李雅普诺夫函数，证明了当$μ>\frac{K_0}{16}$和$K_0=\max\limits_{0\leq v \leq \infty}\frac{|χ(v)|^2}{γ(v)}$同时存在时，\eqref{0-1}的所有解$(u,v)$将指数收敛到唯一的定常稳态$(1,1)$.

英文摘要：

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 (γ(v)\nabla u- uχ(v)\nabla v)+μu(1-u), &x\in Ω, ~~t>0, 0=Δv+ u-v,& x\in Ω, ~~t>0,\\ u(x,0)=u_0(x), & x\in Ω, \end{cases} \end{equation} with homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \R^2$ with smooth boundary, where the motility functions $γ(v)$ and $χ(v)$ satisfy the following conditions \begin{itemize} \item {\color{black}$(γ,χ)\in [C^2[0,\infty)]^2$} with $γ(v)>0$ and {\color{black} $\frac{|χ(v)|^2}{γ(v)}$ is bounded for all $v\geq 0$.} %for all $v\geq 0$ and $\lim\limits_{v\to\infty}\frac{|χ(v)|^2}{γ(v)}$ exists. \end{itemize} By employing the method of energy estimates , we establish the existence of globally bounded solutions of \eqref{0-1} with $μ>0$ for any $u_0 \in W^{1, \infty}(Ω)$. 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 $μ>\frac{K_0}{16}$ with $K_0=\max\limits_{0\leq v \leq \infty}\frac{|χ(v)|^2}{γ(v)}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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