On the Cauchy problem for a two-component b-family system with high order nonlinearity

Abstract

This paper is devoted to the Cauchy problem of two-component b-family system with high order nonlinearity, which includes the well-known Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and their two-component extension systems as special cases. Firstly, we establish the local well-posedness of the system in the non-homogeneous Besov space by using the Littlewood-Paley decomposition and the transport equation theory. Secondly, we investigate the blow-up of strong solutions to the system. Thirdly, we study the Gevrey regularity and analyticity of the solutions to the system in the Gevrey-Sobolev spaces by using the generalized Ovsyannikov theorem. Moreover, we find a lower bound of the lifespan and prove the continuity of the data-to-solution mapping.