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.

中文翻译：

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有界域中趋化性浅水系统的整体存在和长时间行为

在本文中，我们考虑了有界域$ \ Omega \ subset \ mathbb {R} ^ 2 $中的趋化性-浅水系统。通过能量方法，我们建立了具有较小初始扰动的强解的整体存在性，并获得了该解的指数衰减率。我们将有界域分为内部域和直到边界的域。在内部领域，该问题被视为柯西问题。在直到边界的区域中，切线方向和法线方向被不同地对待。我们使用不同的方法来获取切线方向和法线方向的估计值。