Publication Date:
2016-01-10
Description:
This paper is devoted to the attraction–repulsion chemotaxis system with a logistic source: \[ \begin{cases} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\mu\nabla\cdot(u\nabla w)+\mathcal{R}(u),\quad x\in\Omega,\ t 〉 0,\\ \varrho v_t=\Delta v-\alpha_1 v+\beta_1 u,\quad x\in\Omega,\ t 〉 0,\\ \varrho w_t=\Delta w-\alpha_2 w+\beta_2 u,\quad x\in\Omega,\ t 〉 0, \end{cases} \] where $\Omega \subset \mathbb {R}^N(N\geqslant 1)$ is a bounded domain with smooth boundary and $\mathcal {R}(s)\leqslant a-bs^\tau $ . For the case $\varrho =0$ , we show that when the repulsion prevails over the attraction in the sense that $\mu \beta _2-\chi \beta _1 〉 0$ , there exist global bounded classical solutions for any logistic damping $\tau \geqslant 1$ . When the attraction dominates the repulsion in the sense that $\mu \beta _2-\chi \beta _1 〈 0$ , the classical solutions are still global and bounded provided that the logistic damping is strong. For the case $\varrho 〉 0$ , we will investigate the similar problem for $N=1$ and $N=2$ . We will also study the regularity of stationary solutions.
Print ISSN:
0272-4960
Electronic ISSN:
1464-3634
Topics:
Mathematics
