The decay rates of solutions to a chemotaxis-shallow water system

Abstract

In this paper, we consider the large time behavior of solution for the chemotaxis-shallow water system in $\mathbb{R}^2$. The lower bound for time decay rates of the bacterial density and the chemoattractant concentration are proved by the method of energy estimates, which implies these two variables tend to zero at the $L^2$-rate $(1+t)^{-\frac{1}{2}}$. Furthermore, by the Fourier splitting method, we also show the first order spatial derivatives of the bacterial density tends to zero at the $L^2$-rate $(1+t)^{-1}$.

Keywords

• chemotaxis
• shallow water system
• decay rates