Non-uniform continuity of the generalized Camassa-Holm equation in Besov spaces

TL;DR

This paper proves that the solution map for the generalized Camassa-Holm equation is not uniformly continuous in Besov spaces, extending previous results from Sobolev spaces and including the critical space case.

Contribution

It establishes the non-uniform continuity of the solution map in Besov spaces for the generalized Camassa-Holm equation, including the critical space case.

Findings

Solution map not uniformly continuous in Besov spaces

Extension of previous Sobolev space results to Besov spaces

First analysis of non-uniform continuity in the critical space B_{2,1}^{3/2}

Abstract

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.