Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity

Youshan Tao

doi:10.3934/dcdsb.2013.18.2705

Article Contents

2013, Volume 18, Issue 10: 2705-2722. Doi: 10.3934/dcdsb.2013.18.2705

This issue Previous Article Numerical simulation of chemotaxis models on stationary surfaces Next Article

Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity

Youshan Tao^1,

1.
Department of Applied Mathematics, Dong Hua University, Shanghai 200051

Received: October 31, 2012

Revised: February 28, 2013

Published: November 2013

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Abstract

This paper deals with the repulsion chemotaxis system $$ \left\{ \begin{array}{ll} u_t=\Delta u +\nabla \cdot (f(u)\nabla v), & x\in\Omega, \ t>0, \\ v_t=\Delta v +u-v, & x\in\Omega, \ t>0, \end{array} \right. $$ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$ with $n\ge 3$, where $f(u)$ is the density-dependent chemotactic sensitivity function satisfying $$ f \in C^2([0, \infty)), f(0)=0, 0 < f(u) \le K(u+1)^{\alpha} for all u > 0 $$ with some $K>0$ and $\alpha>0$.
It is proved that under the assumptions that $0\not\equiv u_0\in C^0(\bar{\Omega})$ and $v_0\in C^1(\bar{\Omega})$ are nonnegative and that $\alpha<\frac{4}{n+2}$, the classical solutions to the above system are uniformly-in-time bounded. Moreover, it is shown that for any given $(u_0, v_0)$, the corresponding solution $(u, v)$ converges to $(\bar{u}_0, \bar{u}_0)$ as time goes to infinity, where $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.

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Mathematics Subject Classification: Primary: 35A01, 35K51, 35K57, 35M33; Secondary: 35Q92, 92C17.

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