  In this paper, we study a generalized Camassa-Holm (gCH) model with both
dissipation and dispersion, which has (N + 1)-order nonlinearities and includes
the following three integrable equations: the Camassa-Holm, the
Degasperis-Procesi, and the Novikov equations, as its reductions. We first
present the local well-posedness and a precise blow-up scenario of the Cauchy
problem for the gCH equation. Then we provide several sufficient conditions
that guarantee the global existence of the strong solutions to the gCH
equation. Finally, we investigate the propagation speed for the gCH equation
when the initial data is compactly supported.
