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

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 Hölder 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}.