On the Cauchy problem for the generalized Camassa–Holm equation

Abstract

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation, which was proposed by Hakkaev and Kirchev (Comm Part Diff Equ 30:761–781, 2005). Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s},s>3/2\) for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Finally, it is shown that the solution map for the generalized Camassa–Holm equation is Holder continuous in \(H^{r}\)-topology.