Global stabilization of the full attraction-repulsion Keller-Segel system

Hai-Yang Jin; Zhi-An Wang

doi:10.3934/dcds.2020027

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June 2020, 40(6): 3509-3527. doi: 10.3934/dcds.2020027

Global stabilization of the full attraction-repulsion Keller-Segel system

Hai-Yang Jin ^1, and Zhi-An Wang ^2,,

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, China

^* Corresponding author: Zhi-An Wang

To Professor Wei-Ming Ni on the occasion of his 70th birthday, with our best wishes

Received May 2018 Revised March 2019 Published October 2019

Fund Project: The research of H.Y. Jin was supported by the NSF of China No. 11871226, and the Fundamental Research Funds for the Central Universities. The research of Z.A. Wang acknowledges partial supports from an internal grant No. ZZHY in Hong Kong Polytechnic University and from NSFC grant 11571086.

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system

$\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \nabla \cdot (\chi u\nabla v) + \nabla \cdot (\xi u\nabla w),}&{x \in \Omega ,t > 0,}\\{{v_t} = {D_1}\Delta v + \alpha u - \beta v,}&{x \in \Omega ,t > 0,}\\{{w_t} = {D_2}\Delta w + \gamma u - \delta w,}&{x \in \Omega ,t > 0,}\\{u(x,0) = {u_0}(x),v(x,0) = {v_0}(x),w(x,0) = {w_0}(x)}&{x \in \Omega ,}\end{array}} \right.\;\;\;\;\left( * \right)$

in a bounded domain

$ \Omega\subset \mathbb{R}^2 $

with smooth boundary subject to homogeneous Neumann boundary conditions. By constructing an appropriate Lyapunov functions, we establish the boundedness and asymptotical behavior of solutions to the system (*) with large initial data

$ (u_0,v_0,w_0) \in [W^{1,\infty}(\Omega)]^3 $

. Precisely, we show that if the parameters satisfy

$ \frac{\xi\gamma}{\chi\alpha}\geq \max\Big\{\frac{D_1}{D_2},\frac{D_2}{D_1},\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $

for all positive parameters

$ D_1,D_2,\chi,\xi,\alpha,\beta,\gamma $

and

$ \delta $

, the system (*) has a unique global classical solution

$ (u,v,w) $

, which converges to the constant steady state

$ (\bar{u}_0,\frac{\alpha}{\beta}\bar{u}_0,\frac{\gamma}{\delta}\bar{u}_0) $

$ t\to+\infty $

, where

$ \bar{u}_0 = \frac{1}{|\Omega|}\int_\Omega u_0dx $

. Furthermore, the decay rate is exponential if

$ \frac{\xi\gamma}{\chi\alpha}> \max\Big\{\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $

. This paper provides the first results on the full ARKS system with unequal chemical diffusion rates (i.e.

$ D_1\ne D_2 $

) in multi-dimensions.

Keywords: Chemotaxis, attraction-repulsion, global stability, exponential decay rate.

Mathematics Subject Classification: Primary: 35A01, 35B40, 35B44, 35K57, 35Q92, 92C17.

Citation: Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027