Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

Abstract

In this paper we are concerned with a Boussinesq system for smallamplitude long waves arising in nonlinear dispersive media. Considerations will be given for the global well-posedness and the time decay rates of solutions when the model is posed on a periodic domain and a general class of damping operator acts 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, the result is extended for the full system.

Long-time asymptotics of a linear higher-order water wave model

“…Results of stabilization for the higher-order system (4) are obtained in [3] when a 1 = c 1 = 0 and general damping is considered in each equation. Controllability results for system (4) are studied in [1] when a control localized in the interior of the domain is considered and acts on only one equation.…”

On the controllability of a model system for long waves in nonlinear dispersive media

“…They 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. More recently, a higher-order Boussinesq systems of BBM-BBM type (a = a 1 = c = c 1 = 0) was considered in [2]. The global well-posedness and the time decay rates of solutions were studied when the model is posed on a periodic domain and a general class of damping operator acts in each equation.…”

A note on the control and stabilization of a higher-order water wave model

“…where θ ∈ [0, 1]. The numbers r 0 and q 0 of boundary conditions in (2) depend on the values of the parameters of the system. For instance, if a 1 = b 1 = 0 and a ̸ = 0, then r 0 = 2.…”

“…However, the results available in the literature do not give an immediate answer to the problems addressed here. In this sense, we refer to [1,2] for a quite complete review of the field. The remainder of this paper is organized as follows: in Section 2, we introduce some notation and the well-posedness of the linearized system.…”