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

Abstract

This paper deals with a two-competing-species chemotaxis-fluid system with two different signals $$\begin{aligned} \left\{ \begin{aligned}&(n_{1})_{t}+\mathbf{u }\cdot \nabla n_{1}=d_{1}\Delta n_{1}-\chi _{1}\nabla \cdot (n_{1}\nabla c) +\mu _{1} n_{1}(1-n_{1}-a_{1}n_{2}),&\text {in}\; \Omega \times (0,\infty ), \\c \Omega \times (0,\infty ), \\&(n_{2})_{t}+\mathbf{u }\cdot \nabla n_{2}=d_{3}\Delta n_{2}-\chi _{2}\nabla \cdot (n_{2}\nabla v) +\mu _{2}n_{2}(1-a_{2}n_{1}-n_{2}),&\text {in}\; \Omega \times (0,\infty ), \\v \Omega \times (0,\infty ), \\&\mathbf{u }_{t}+\kappa (\mathbf{u }\cdot \nabla )\mathbf{u }=\Delta \mathbf{u } +\nabla P+(\beta _{1} n_{1}+\beta _{2} n_{2})\nabla \phi ,&\text {in}\; \Omega \times (0,\infty ), \\&\nabla \cdot \mathbf{u }=0,&\text {in}\; \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$ in a smooth bounded domain $$\Omega \subset {\mathbb {R}}^{N}$$ , $$N=2,3$$ , under homogeneous Neumann boundary conditions for $$n_{1}, n_{2}, c, v$$ and zero Dirichlet boundary condition for $$\mathbf{u }$$ , where $$\kappa \in \{0,1\}$$ , the parameters $$d_{i}$$ ( $$i=1,2,3,4$$ ) and $$\chi _{j},\mu _{j}, a_{j}, \alpha _{j},\beta _{j}$$ ( $$j=1,2$$ ) are positive. This system describes the evolution of two-competing species which react on two different chemical signals in a liquid surrounding environment. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. Firstly, when $$N=2$$ and $$\kappa =1$$ , based on the standard heat-semigroup argument, it is proved that for all appropriately regular nonnegative initial data and any positive parameters, this system possesses a unique global bounded solution. Secondly, when $$N=3$$ and $$\kappa =0$$ , by using the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that there exists $$\theta _{0}>0$$ such that $$\frac{\max \{\chi _{1},\chi _{2}\}}{\min \{\mu _{1},\mu _{2}\}}<\theta _{0}$$ . Finally, by means of energy functionals and comparison arguments, it is shown that the global bounded solution of the system converges to different constant steady states, according to the different values of $$a_{1}$$ and $$a_{2}$$ . Furthermore, we give the precise convergence rates of global solutions.