# American Institute of Mathematical Sciences

September  2020, 40(9): 5415-5439. doi: 10.3934/dcds.2020233

## Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion

 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: jinchhua@126.com

Received  November 2019 Revised  March 2020 Published  June 2020

Fund Project: This work is supported by NSFC(11871230), Guangdong Basic and Applied Basic Research Foundation(2020B1515310013)

This paper is concerned with the time periodic problem to a coupled chemotaxis-fluid model with porous medium diffusion $\Delta n^m$. The global existence of solutios for the initial and boundary value problem of this model have been studied by many authors, and in particular, the global solvability is established for $m>\frac65$ in dimension 3. Here, taking advantage of a double-level approximation scheme, we establish the existence of uniformly bounded time periodic solution for any $m\ge \frac 65$ and any large periodic source $g(x, t)$. In particular, the energy estimates techniques we used also applicable to the proof of global existence of the initial-boundary value problem, and one can supply the existence of global solutions for $m = \frac65$ by this method.

Citation: Jiapeng Huang, Chunhua Jin. Time periodic solution to a coupled chemotaxis-fluid model with porous medium diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5415-5439. doi: 10.3934/dcds.2020233
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