Electron. J. Differential Equations, Vol. 2017 (2017), No. 254, pp. 1-9.

Existence and asymptotic behavior of global solutions to chemorepulsion systems with nonlinear sensitivity

Yulin Lai, Youjun Xiao

Abstract:
This article concerns the chemorepulsion system with nonlinear sensitivity and nonlinear secretion
$$\displaylines{ u_t=\Delta u+\nabla\cdot(\chi u^m\nabla v),\quad x\in\Omega,\; t>0,\cr 0=\Delta v-v+u^\alpha,\quad x\in\Omega,\; t>0, }$$
under homogeneous Neumann boundary conditions, where $\chi>0$, m>0, $\alpha>0$, $\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary. The existence and uniform boundedness of a classical global solutions are obtained. Furthermore, it is shown that for any given $u_0$, if $\alpha>m$ or $\alpha\ge 1$, the corresponding solution (u,v) converges to $(\bar{u}_0,\bar{u}^\alpha_0)$ as time goes to infinity, where $\bar{u}_0:=\frac1{|\Omega|}\int_\Omega u_0dx$.

Submitted April 7, 2017. Published October 10, 2017.
Math Subject Classifications: 35K55, 35Q92, 35Q35, 92C17.
Key Words: Chemotaxis; repulsion; nonlinear sensitivity;global solution; asymptotic behavior.

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Yulin Lai
Third People's Hospital of Yibin City
Yibin 644000, China
email: 32212779@qq.com
Youjun Xiao
College of Mathematic & Information
China West Normal University
Nanchong 637002, China
email: mathxyj@126.com

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