Sharp ill-posedness for the generalized Camassa–Holm equation in Besov spaces

Abstract

In this paper, we consider the Cauchy problem for the generalized Camassa–Holm equation that containing, as its members, three integrable equations: the Camassa–Holm equation, the Degasperis–Procesi equation and the Novikov equation. We present a new and unified method to prove the sharp ill-posedness for the generalized Camassa–Holm equation in $$B^s_{p,\infty }$$ with $$s>\max \{1+1/p, 3/2\}$$ and $$1\le p\le \infty $$ in the sense that the solution map to this equation starting from $$u_0$$ is discontinuous at $$t = 0$$ in the metric of $$B^s_{p,\infty }$$ . Our result covers and improves the previous work given in Li et al. (J Differ Equ 306:403–417, 2022), solving an open problem left in Li et al. (2022).