Stabilization of a Boussinesq system with generalized damping

Abstract

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin–Bona–Mahony with periodic boundary conditions. We study the stability properties of the resulting system when generalized damping operators are introduced in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, we show that the same property holds for the nonlinear system.