| Abstract: |
| In this talk, we consider a two-competing-species chemotaxis-fluid system with two different signals under homogeneous Neumann boundary conditions in a smooth bounded domain. 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 under some suitable conditions. 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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