Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain

Weike Wang; Yucheng Wang

doi:10.3934/dcds.2020284

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2020, Volume 40, Issue 11: 6379-6409. Doi: 10.3934/dcds.2020284

This issue Previous Article Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations Next Article Identifying varying magnetic anomalies using geomagnetic monitoring

Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain

Weike Wang^1, and
Yucheng Wang^2, ,

1.
School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

^* Corresponding author: Yucheng Wang
^* Corresponding author: Yucheng Wang

Received: January 2020

Revised: June 2020

Published: July 2020

The authors are supported by the National Natural Science Foundation of China (No. 11771284 and No. 11831011)

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Abstract

In this paper, we consider the chemotaxis–shallow water system in a bounded domain $ \Omega\subset\mathbb{R}^2 $. By energy method, we establish the global existence of strong solution with small initial perturbation and obtain the exponential decaying rate of the solution. We divide the bounded domain into interior domain and the domain up to the boundary. In the interior domain, the problem is treated like the Cauchy problem. In the domain up to the boundary, the tangential and normal directions are treated differently. We use different method to get the estimates for the tangential and normal directions.

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Mathematics Subject Classification: Primary: 35Q35, 35Q92, 92C17, 76N10.

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