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

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Abstract

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

$$\begin{aligned} \left\{ \begin{aligned}&(n_{1})_{t}+\mathbf{u }\cdot \nabla n_{1}=d_{1}\Delta n_{1}-\chi _{1}\nabla \cdot (n_{1}\nabla c) +\mu _{1} n_{1}(1-n_{1}-a_{1}n_{2}),&\text {in}\; \Omega \times (0,\infty ), \\&c_{t}+\mathbf{u }\cdot \nabla c=d_{2}\Delta c-\alpha _{1} cn_{2},&\text {in}\; \Omega \times (0,\infty ), \\&(n_{2})_{t}+\mathbf{u }\cdot \nabla n_{2}=d_{3}\Delta n_{2}-\chi _{2}\nabla \cdot (n_{2}\nabla v) +\mu _{2}n_{2}(1-a_{2}n_{1}-n_{2}),&\text {in}\; \Omega \times (0,\infty ), \\&v_{t}+\mathbf{u }\cdot \nabla v=d_{4}\Delta v-\alpha _{2}vn_{1},&\text {in}\; \Omega \times (0,\infty ), \\&\mathbf{u }_{t}+\kappa (\mathbf{u }\cdot \nabla )\mathbf{u }=\Delta \mathbf{u } +\nabla P+(\beta _{1} n_{1}+\beta _{2} n_{2})\nabla \phi ,&\text {in}\; \Omega \times (0,\infty ), \\&\nabla \cdot \mathbf{u }=0,&\text {in}\; \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

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.

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Acknowledgements

The authors would like to deeply thank the editor and reviewers for their insightful and constructive comments. Pan Zheng is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No: cstc2019jcyj-msxmX0082). Robert Willie is partially supported by UKZN-Cost Center: RC43-621994-2006-YY. Chunlai Mu is partially supported by National Natural Science Foundation of China (Grant No: 11771062) and the Fundamental Research Funds for the Central Universities, China (Grant No: 10611CDJXZ238826).

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Zheng, P., Willie, R. & Mu, C. Global Boundedness and Stabilization in a Two-Competing-Species Chemotaxis-Fluid System with Two Chemicals. J Dyn Diff Equat 32, 1371–1399 (2020). https://doi.org/10.1007/s10884-019-09797-4

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