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

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 boundedconvex domain $\Omega\subset\mathbb{R}^n$ with $n\ge 3$, where $f(u)$ is thedensity-dependent chemotactic sensitivity function satisfying$$f \in C^2([0, \infty)), f(0)=0, 0 0$$with some $K>0$ and $\alpha>0$. It is proved that under the assumptions that $0\not\equiv u_0\inC^0(\bar{\Omega})$ and $v_0\in C^1(\bar{\Omega})$ are nonnegativeand that $\alpha<\frac{4}{n+2}$, the classical solutions to theabove system are uniformly-in-time bounded. Moreover, it is shownthat 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 $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.