Global Boundedness in a Two-Competing-Species Chemotaxis System with Two Chemicals

Abstract

This paper deals with a two-competing-species chemotaxis system with two different chemicals

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla v)+\mu_{1} u(1-u-a _{1}w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau v_{t}=\Delta v-v+w, & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle w_{t}=\Delta w-\chi_{2}\nabla \cdot (w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau z_{t}=\Delta z-z+u, & (x,t)\in \varOmega \times (0,\infty ), \end{array}\displaystyle \right . \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\varOmega \subset \mathbb{R}^{n}$ $(n\geq 1)$ with the nonnegative initial data $(u_{0},\tau v_{0},w_{0},\tau z_{0})\in C^{0}(\overline{\varOmega }) \times W^{1,\infty }(\varOmega )\times C^{0}(\overline{\varOmega })\times W ^{1,\infty }(\varOmega )$, where $\tau \in \{0,1\}$ and the parameters $\chi_{i},\mu_{i},a_{i}$ ($i=1,2$) are positive. When $\tau =0$, based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, the system possesses a unique globally bounded classical solution for any positive parameters if $n=2$. On the other hand, when $\tau =1$, relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that $n\geq 1$ and there exists $\theta_{0}>0$ such that $\frac{\chi_{2}}{ \mu_{1}}<\theta_{0}$ and $\frac{\chi_{1}}{\mu_{2}}<\theta_{0}$.