Global boundedness and asymptotic behavior of the solutions to an attraction–repulsion chemotaxis-growth system

Abstract

This paper deals with the following attraction–repulsion chemotaxis system with logistic source $\{\begin{array}{ll}{\displaystyle u_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)+\xi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)+f(u),}& x\in \mathrm{\Omega },t>0,\\ {\displaystyle 0=\mathrm{\Delta }v+\alpha u-\beta v,}& x\in \mathrm{\Omega },t>0,\\ {\displaystyle w_t=\mathrm{\Delta }w+\gamma u-\delta w,}& x\in \mathrm{\Omega },t>0\end{array}$under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$ with smooth boundary, where $\chi ,\xi ,\alpha ,\beta ,\gamma ,\delta$ are assumed to be positive constants and $f(s)=\mu s(1-s^{\theta })$ with $\mu >0$ and $\theta \ge 1$. It is shown that the system admits a unique globally bounded classical solution provided that space dimension n = 2, or $n\ge 3$ and $\theta >1$, or $n\ge 3,\theta =1$ and $\mu >C(n)\xi \gamma +\chi \alpha$ with some $C(n)>0$. Furthermore, under the additional assumption μ is suitably large, we show that the global classical solution will converge to the constant steady state $(1,\frac{\alpha }{\beta },\frac{\gamma }{\delta })$ exponentially as $t\to \mathrm{\infty }$. Our results imply that the logistic source plays an important role on the behavior of the solutions in this model.

Keywords:

• Attraction–repulsion
• boundedness
• logistic source
• large time behavior
• chemotaxis

AMS (2010) Subject Classifications:

• 35A23
• 35K90
• 92C17
• 35B40
• 35Q92

Acknowledgments

The paper is supported by the National Science Foundation of China(11301419) and the Meritocracy Research Funds of China West Normal University [17YC382].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11301419] and Meritocracy Research Funds of China West Normal University [grant number 17YC382].