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

Jiapeng Huang; Chunhua Jin

doi:10.3934/dcds.2020233

Article Contents

2020, Volume 40, Issue 9: 5415-5439. Doi: 10.3934/dcds.2020233

This issue Previous Article Super fast vanishing solutions of the fast diffusion equation Next Article Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure

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

Jiapeng Huang and
Chunhua Jin^,

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

^* Corresponding author: jinchhua@126.com
^* Corresponding author: jinchhua@126.com

Received: October 31, 2019

Revised: February 29, 2020

Published: June 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B10, 92C17; Secondary: 35M10, 35D30.

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