On the well posedness and large-time behavior of higher order Boussinesq system

TL;DR

This paper studies the mathematical properties and stabilization of advanced Boussinesq systems modeling shallow water waves, establishing well-posedness and exponential decay of solutions through feedback control.

Contribution

It introduces a new family of feedback laws ensuring well-posedness and exponential stability for higher order Boussinesq systems on bounded domains.

Findings

System solutions are locally well-posed under the proposed feedback laws.

Linearized system solutions decay exponentially over time.

The approach applies to a family of Korteweg-de Vries--type Boussinesq systems.

Abstract

A family of Boussinesq systems has been proposed to describe the bi-directional propagation of small amplitude long waves on the surface of shallow water. In this paper, we investigate the well-posedness and boundary stabilization of the generalized higher order Boussinesq systems of Korteweg-de Vries--type posed on a interval. We design a two-parameter family of feedback laws for which the system is locally well-posed and the solutions of the linearized system are exponentially decreasing in time.