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doi: 10.3934/dcdsb.2021193
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## A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity

 College of Mathematics and Information, China West Normal University, NanChong 637000, China

* Corresponding author: Jie Zhao

Received  November 2020 Revised  June 2021 Early access July 2021

This paper deals with the dynamical properties of the quasilinear parabolic-parabolic chemotaxis system
 $\begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(\frac{u}{v} \nabla v)+\mu u- \mu u^{2}, \, \, \, &x\in\Omega, \, \, \, t>0, \\ v_{t} = \Delta v-v+u, &x\in\Omega, \, \, \, t>0, \end{array} \right. \end{eqnarray*}$
under homogeneous Neumann boundary conditions in a convex bounded domain
 $\Omega\subset\mathbb{R}^{n}$
,
 $n\geq2$
, with smooth boundary.
 $\chi>0$
and
 $\mu>0$
,
 $D(u)$
is supposed to satisfy the behind properties
 $\begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*}$
It is shown that there is a positive constant
 $m_{*}$
such that
 $\begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*}$
for all
 $t\geq0$
. Moreover, we prove that the solution is globally bounded. Finally, it is asserted that the solution exponentially converges to the constant stationary solution
 $(1, 1)$
.
Citation: Jie Zhao. A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021193
##### References:

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