Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop

Abstract

This paper deals with the two-species chemotaxis-competition system with loop

$$\begin{aligned} \left\{ \begin{array}{llll} \partial _{t} u_{1}=d_1\Delta u_{1}-\chi _{11}\nabla \cdot (u_{1}\nabla v_{1}) -\chi _{12}\nabla \cdot (u_{1}\nabla v_{2}) +\mu _{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ \partial _{t} u_{2}=d_2\Delta u_{2}-\chi _{21}\nabla \cdot (u_{2}\nabla v_{1}) -\chi _{22}\nabla \cdot (u_{2}\nabla v_{2}) +\mu _{2}u_{2}(1-u_{2}-a_{2}u_{1}),\\ \partial _t v_1=d_3\Delta v_{1}-\lambda _{1} v_{1}+\alpha _{11}u_{1}+\alpha _{12}u_{2},\\ \partial _t v_2=d_4\Delta v_{2}-\lambda _{2} v_{2}+\alpha _{21}u_{1}+\alpha _{22}u_{2}, \\ \end{array} \right. \end{aligned}$$

subject to homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb {R}}^{3}$, where $\chi _{ij}>0$, $\mu _{i}>0$, $a_i>0$, $\alpha _{ij}>0$, $\lambda _{i}>0$, $d_k>0$ $(i, j=1, 2, k=1, 2, 3, 4)$. Our main purpose is to extend the global boundedness result to the 3D setting. To address this issue, based on a new coupled function, by selecting sufficiently large $\mu _1$ and $\mu _2$, we construct a Gronwall type inequality which directly renders the uniform boundedness of solutions.