On the Cauchy problem for a Camassa-Holm type equation with both dissipation and dispersion

Abstract In this paper, we consider the equation of Camassa-Holm type with both dissipation and dispersion, which has (λ + 2)-order nonlinearities and includes the following three integrable equations: the Camassa-Holm, the Degasperis-Procesi, and the Novikov equations. Firstly, we prove the local well-posedness of this equation by Kato’s Theorem and obtain the blow-up phenomena of the corresponding Cauchy problem. Then, several sufficient conditions are derived to guarantee the global existence of the strong solutions.