Boundedness in a three-dimensional two-species chemotaxis system with two chemicals

Abstract

This paper deals with a two-species chemotaxis system with Lotka–Volterra competitive kinetics

$$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\chi _1\nabla \cdot (u\nabla v)+\mu _1u(1-u-a_1w),&{}\quad x\in \Omega ,\quad t>0,\\ v_t=\Delta v-v+w,&{}\quad x\in \Omega ,\quad t>0,\\ w_t=\Delta w-\chi _2\nabla \cdot (w\nabla z)+\mu _2w(1-w-a_2u),&{}\quad x\in \Omega ,\quad t>0,\\ z_t=\Delta z-z+u,&{}\quad x\in \Omega ,\quad t>0, \end{array} \right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^3$ is a bounded domain with smooth boundary $\partial \Omega$; the parameters $\chi _i,\,\,\mu _i\,\,\text {and}\,\,a_i$ $(i = 1,2)$ are positive, and the nonnegative initial data $(u_{0}, v_{0},w_{0}, z_{0})\in C^{0}({\overline{\Omega }})\times W^{1,\infty }(\Omega )\times C^{0}({\overline{\Omega }})\times W^{1, \infty }(\Omega )$. It is shown that the system possesses a unique global bounded classical solution provided that $\mu _i (i=1,2)$ are sufficiently large.