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

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 $\bar{u}_0:=\frac1{|\Omega|}\int_\Omega u_0dx$