论文标题
具有逻辑生长和信号依赖性运动的凯勒 - 塞格系统
The Keller-Segel system with logistic growth and signal-dependent motility
论文作者
论文摘要
本文与以下具有非线性运动功能的趋化系统有关\ begin {equation} \ label {0-1} \ tag {$ \ ast $} \ begin {cases} u_t = \ nabla \ cdot（γ（γ（v）\ nabla u-u- u-（v） ~~ t> 0， 0 =ΔV+ u-v，＆x \ inω，~~ t> 0，\\ u（x，0）= u_0（x），＆x \ inω，\ end end {cases} \ end {cases} \ end {equeation}，带有均质的neumann neumann neumann边界条件，在有界的域$ω\ subset \ r^2 $ $^2 $的情况下，功能和功能。 $χ（v）$满足以下条件\ begin {inatize} \ item {\ color {black} $（γ，χ）\ in [c^2 [0，\ infty）]^2 $}，用$γ（v）> 0 $ $ unding $ $ and {\ color {black} $ {black} $ \ frac inding（ $ v \ geq 0 $。}％对于所有$ v \ geq 0 $和$ \ lim \ limits_ {v \ to \ infty} \ frac {|χ（v）|^2} {γ（v）} $已存在。 \ end {inatizize}通过采用能量估计方法，我们建立了\ eqref {0-1}的全球限制解决方案的存在，并使用$μ> 0 $，用于任何$ u_0 \ in W^{1，\ infty}中的任何$ u_0 \（ω）（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）} $。
The paper is concerned with the following chemotaxis system with nonlinear motility functions \begin{equation}\label{0-1}\tag{$\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 $K_0=\max\limits_{0\leq v \leq \infty}\frac{|χ(v)|^2}{γ(v)}$.
