Non-uniform dependence on initial data for the generalized Camassa-Holm equation in C¹

It is shown in [Adv. Differ. Equ(2017)]{HT} that the Cauchy problem for the generalized Camassa-Holm equation is well-posed in C¹ and the data-to-solution map is H\"{o}lder continuous from C^α to C([0,T];C^α) with α∈[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 C¹. In particular, our result also can be a complement of previous work on the classical Camassa-Holm equation in [Geom. Funct. Anal(2002)]{G02}.