# American Institute of Mathematical Sciences

March  2020, 40(3): 1737-1755. doi: 10.3934/dcds.2020091

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

 College of Mathematics and Information, China West Normal University, Nanchong 637009, China

Received  May 2019 Revised  October 2019 Published  December 2019

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
 $(1, 1)$
.
Citation: Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091
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