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June  2021, 26(6): 3023-3041. doi: 10.3934/dcdsb.2020218

## The Keller-Segel system with logistic growth and signal-dependent motility

 1 Department of Mathematics, South China University of Technology, Guangzhou, 510640, China 2 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong

*Corresponding author: Zhi-An Wang

Received  February 2020 Revised  May 2020 Published  July 2020

Fund Project: The research of H.Y. Jin was supported by the NSF of China (No. 11871226), Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515010140), Guangzhou Science and Technology Program No.202002030363 and the Fundamental Research Funds for the Central Universities. The research of Z.A. Wang was supported by the Hong Kong RGC GRF grant 15303019 (Project ID P0030816)

The paper is concerned with the following chemotaxis system with nonlinear motility functions
 $$$\label{0-1}\begin{cases}u_t = \nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-u), &x\in \Omega, ~~t>0, \\ 0 = \Delta v+ u-v, & x\in \Omega, ~~t>0, \\u(x, 0) = u_0(x), & x\in \Omega, \end{cases}~~~~(\ast)$$$
subject to homogeneous Neumann boundary conditions in a bounded domain
 $\Omega\subset \mathbb{R}^2$
with smooth boundary, where the motility functions
 $\gamma(v)$
and
 $\chi(v)$
satisfy the following conditions
 $(\gamma, \chi)\in [C^2[0, \infty)]^2$
with
 $\gamma(v)>0$
and
 $\frac{|\chi(v)|^2}{\gamma(v)}$
is bounded for all
 $v\geq 0$
.
By employing the method of energy estimates, we establish the existence of globally bounded solutions of ($\ast$) with
 $\mu>0$
for any
 $u_0 \in W^{1, \infty}(\Omega)$
with
 $u_0 \geq (\not\equiv) 0$
. Then based on a Lyapunov function, we show that all solutions
 $(u, v)$
of ($\ast$) will exponentially converge to the unique constant steady state
 $(1, 1)$
provided
 $\mu>\frac{K_0}{16}$
with
 $K_0 = \max\limits_{0\leq v \leq \infty}\frac{|\chi(v)|^2}{\gamma(v)}$
.
Citation: Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218
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