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.

本文处理以下具有逻辑源的吸引-排斥趋化系统$\{\begin{array}{ll}{\displaystyle {你}_{吨}=\mathrm{\Delta }你-\chi \mathrm{\nabla }\cdot (你\mathrm{\nabla }v)+\xi \mathrm{\nabla }\cdot (你\mathrm{\nabla }w)+F(你),}& X\in \mathrm{\Omega },吨>0,\\ {\displaystyle 0=\mathrm{\Delta }v+\alpha 你-\beta v,}& X\in \mathrm{\Omega },吨>0,\\ {\displaystyle w_{吨}=\mathrm{\Delta }w+\gamma 你-\delta w,}& X\in \mathrm{\Omega },吨>0\end{array}$在有界域中的齐次 Neumann 边界条件下$\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$具有平滑边界，其中$\chi ,\xi ,\alpha ,\beta ,\gamma ,\delta$假定为正常数，并且$F(s)=\mu s(1-s^{\theta })$和$\mu >0$和$\theta \ge 1$. 结果表明，如果空间维数n  = 2，则系统承认一个唯一的全局有界经典解，或$n\ge 3$和$\theta >1$， 或者$n\ge 3,\theta =1$和$\mu >C(n)\xi \gamma +\chi \alpha$和一些$C(n)>0$. 此外，在附加假设μ适当大的情况下，我们表明全局经典解将收敛到恒定稳态$(1,\frac{\alpha }{\beta },\frac{\gamma }{\delta })$指数地为$吨\to \mathrm{\infty }$. 我们的结果表明，逻辑源对该模型中解决方案的行为起着重要作用。