In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation proposed by Hakkaev and Kirchev (2005) \cite{ Hakkaev 2005}. We prove that the solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result include the present work (2020) \cite{ Li 2020,Li 2020-1} on Camassa-Holm equation with $Q=1$ and extends the previous non-uniform continuity in Sobolev spaces (2015) \cite{ Mi 2015} to Besov spaces. In addition, the non-uniform continuity in critical space $B_{ 2, 1}^{ \frac{ 3}{ 2}}(\mathbb{ R})$ is the first to be considered in our paper.
Categories: math.AP