Wave breaking and global existence for a family of peakon equations with high order nonlinearity

Abstract

This paper is concerned with the Cauchy problem for a family of peakon equations with high order nonlinearity. To begin with, the local well-posedness results in Besov and Sobolev spaces are discussed. Then we obtain the precise blow-up scenario for strong solutions to the equation, which is conjectured by Anco et al., (2015). Moreover, we establish some global existence results for the strong solutions by deriving two useful conservation laws. Furthermore, wave breaking phenomenon is studied and we construct a new finite time blow-up solution to the equation with respect to the initial data.