Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

Abstract

This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity

$$\left\{\begin{array}{cc}{u}_t=\Delta u-\chi \nabla \cdot \left(u\nabla lnv\right)+ru-\mu u^2,\hfill & (x,t)\in \Omega \times (0,\infty ),\hfill \\ v_t=\Delta v-v+u,\hfill & (x,t)\in \Omega \times (0,\infty ),\hfill \end{array}\right.$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb{R}}^2$, where the parameters $\chi ,\mu >0$ and $r\in \mathbb{R}$. Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.