Non-uniform dependence on initial data for the generalized Camassa-Holm equation in $C^1$

TL;DR

This paper demonstrates that the solution map for the generalized Camassa-Holm equation in $C^1$ is not uniformly continuous, highlighting nuanced dependence on initial data beyond previous well-posedness results.

Contribution

It establishes the non-uniform dependence of the solution map in $C^1$, extending understanding of the equation's sensitivity to initial conditions.

Findings

Solution map is not uniformly continuous in $C^1$.

Complements previous work on classical Camassa-Holm equation.

Highlights nuanced dependence on initial data.

Abstract

It is shown in \cite[Adv. Differ. Equ(2017)]{HT} that the Cauchy problem for the generalized Camassa-Holm equation is well-posed in $C^1$ and the data-to-solution map is H\"{o}lder continuous from $C^\alpha$ to $\mathcal{C}([0,T];C^\alpha)$ with $\alpha\in[0,1)$. In this paper, we further show that the data-to-solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in $C^1$. In particular, our result also can be a complement of previous work on the classical Camassa-Holm equation in \cite[Geom. Funct. Anal(2002)]{G02}.