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- W3026190572 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- uchi(v)nabla v)+mu u(1-u), &xin Omega, ~~t>0, 0=Delta v+ u-v,& xin Omega, ~~t>0, u(x,0)=u_0(x), & xin Omega, end{cases} end{equation} with homogeneous Neumann boundary conditions in a bounded domain $Omegasubset R^2$ 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 $vgeq 0$.} %for all $vgeq 0$ and $limlimits_{vtoinfty}frac{|chi(v)|^2}{gamma(v)}$ 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 $K_0=maxlimits_{0leq v leq infty}frac{|chi(v)|^2}{gamma(v)}$." @default.
- W3026190572 created "2020-05-29" @default.
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- W3026190572 date "2020-05-23" @default.
- W3026190572 modified "2023-09-27" @default.
- W3026190572 title "The Keller-Segel system with logistic growth and signal-dependent motility" @default.
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- W3026190572 doi "https://doi.org/10.48550/arxiv.2005.11462" @default.
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