Electronic Journal of Differential Equations (Oct 2017)
Existence and asymptotic behavior of global solutions to chemorepulsion systems with nonlinear sensitivity
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 $\bar{u}_0:=\frac1{|\Omega|}\int_\Omega u_0dx$.