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

In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation that includes the Camassa-Holm as well as the Novikov equation on the line. 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 $1\leq p\leq\infty$ 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 results cover and improve the previous work given in [J. Li, Y. Yu, W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differential Equations, 306 (2022), 403--417], solving an open problem left in [J. Li, Y. Yu, W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differential Equations, 306 (2022), 403--417].