Large time behavior of solution to quasilinear chemotaxis system with logistic source

This paper deals with the quasilinear parabolic-elliptic chemotaxis system

$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u \nabla v)+\mu u- \mu u^{r}, \, \, \, &x\in\Omega, \, \, \, t>0, \\ \tau v_{t} = \Delta v-v+u, &x\in\Omega, \, \, \, t>0, \end{array} \right. \end{eqnarray*} $

under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\mathbb{R}^{n} $ with smooth boundary, where $ \tau\in\{0, 1\} $, $ \chi>0 $, $ \mu>0 $ and $ r\geq2 $. $ D(u) $ is supposed to satisfy

$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $

It is shown that when $ \mu>\frac{\chi^{2}}{16} $ and $ r\geq2 $, then the solution to the system exponentially converges to the constant stationary solution $ (1, 1) $.