Wave breaking for shallow water models with time decaying solutions

Abstract

A family of Camassa-Holm type equations with a linear term and cubic and quartic nonlinearities is considered. Local well-posedness results are established via Kato's approach. Conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is time-dependent. If such coefficient is positive, then the energy functional is a monotonically decreasing function of time, bounded from above by the Sobolev norm of the initial data, and all solutions of the equation are dissipative. Sufficient conditions for the global existence of solutions are described. The existence of wave breaking phenomena is also investigated and necessary conditions for its existence are obtained. In our framework the wave breaking is guaranteed, among other conditions, when the coefficient of the linear term is sufficiently small, which allows us to interpret the equation as a linear perturbation of some recent Camassa-Holm type equations considered in the literature. 2010 AMS Mathematics Classification numbers: 35A01, 35L65, 37K05.