  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 a time-dependent, monotonically decreasing function of time, and bounded
from above by the Sobolev norm of the initial data under some conditions. The
existence of wave breaking phenomenon is 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.
