Electron. J. Diff. Equ., Vol. 2016 (2016), No. 176, pp. 1-21.

Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source

Yilong Wang

Abstract:
This article concerns the attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source,
$$\displaylines{ u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla\cdot(\chi u\nabla v) +\nabla\cdot(\xi u\nabla w)+ru-\mu u^\eta, \cr x\in\Omega,\; t>0,\cr v_t=\Delta v+\alpha u-\beta v, \quad x\in\Omega, \; t>0,\cr w_t=\Delta w+\gamma u-\delta w, \quad x\in\Omega,\; t>0 }$$
under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary. We show that if the diffusion is strong enough or the logistic dampening is sufficiently powerful, then the corresponding system possesses a global bounded classical solution for any sufficiently regular initial data. Moreover, it is proved that if $r=0$, $\beta>\frac{1}{2(\eta-1)}$ and $\delta>\frac{1}{2(\eta-1)}$ for the latter case, then $u(\cdot,t)\to 0$, $ v(\cdot,t)\to 0$ and $ w(\cdot,t)\to 0$ in $L^\infty(\Omega)$ as $ t \to \infty$.

Submitted February 8, 2016. Published July 6, 2016.
Math Subject Classifications: 35K55, 35Q35, 35Q92, 92C17.
Key Words: Chemotaxis; attraction-repulsion; boundedness; nonlinear diffusion; logistic source.

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Yilong Wang
School of Sciences
Southwest Petroleum University
Chengdu 610500, China
email: wangelongelone@163.com

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