A Note on a Generalized Two Component Camassa-Holm System in Sobolev Spaces

Abstract

In this paper, we consider a generalized two component Camassa-Holm system. Based on local well-posedness results and lifespan estimates, we establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from product Sobolev spaces $H^s\times H^s$ to $C([0,T];H^s\times H^s)$. The proof of nonuniform dependence is based upon approximate solutions.