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A family of Camassa-Holm type equations with a linear term and cubic andquartic nonlinearities is considered. Local well-posedness results areestablished via Kato's approach. Conserved quantities for the equation aredetermined and from them we prove that the energy functional of the solutionsis a time-dependent, monotonically decreasing function of time, and boundedfrom above by the Sobolev norm of the initial data under some conditions. Theexistence of wave breaking phenomenon is investigated and necessary conditionsfor its existence are obtained. In our framework the wave breaking isguaranteed, among other conditions, when the coefficient of the linear term issufficiently small, which allows us to interpret the equation as a linearperturbation of some recent Camassa-Holm type equations considered in theliterature.

Wave breaking for shallow water models with time decaying solutions