Electron. J. Diff. Equ., Vol. 2015 (2015), No. 146, pp. 1-13.

Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system

Yuhuan Li, Ke Lin, Chunlai Mu

Abstract:
This article is concerned with the model
$$\displaylines{ u_t=\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\quad x\in \Omega,\; t>0,\cr 0=\Delta v+\alpha u-\beta v,\quad x\in\Omega,\; t>0,\cr 0=\Delta w+\gamma u-\delta w,\quad x\in\Omega,\; t>0 }$$
with homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset \mathbb{R}^{n}\;(n=2,3)$. Under the critical condition $\chi \alpha-\xi \gamma=0$, we show that the system possesses a unique global solution that is uniformly bounded in time. Moreover, when $n=2$, by some appropriate smallness conditions on the initial data, we assert that this solution converges to ($\bar{u}_0$, $\frac{\alpha}{\beta}\bar{u}_0$, $\frac{\gamma}{\delta}\bar{u}_0$) exponentially, where $\bar{u}_0:=\frac{1}{|\Omega|}\int_{\Omega}u_0$.

Submitted April 17, 2015. Published June 6, 2015.
Math Subject Classifications: 35A01, 35B40, 35K55, 92C17.
Key Words: Chemotaxis; attraction-repulsion; boundedness; convergence.

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Yuhuan Li
Department of Mathematics
Sichuan Normal University
Chengdu 610066, China
email: liyuhuanhuan@163.com
Ke Lin
College of Mathematics and Statistics
Chongqing University
Chongqing 401331, China
email: shuxuelk@126.com
Chunlai Mu
College of Mathematics and Statistics
Chongqing University
Chongqing 401331, China
email: clmu2005@163.com

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