Ill-posedness for a generalized Camassa-Holm equation with higher-order nonlinearity in the critical Besov space

In this paper, we prove that the Cauchy problem for a generalized Camassa-Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space $B_{\mathrm{\infty },1}^1(\mathbb{R})$. It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa-Holm equation is ill-posed in $B_{\mathrm{\infty },1}^1(\mathbb{R})$, here we turn our attention to a higher-order nonlinear generalization of Camassa-Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we get the norm inflation in the critical space $B_{\mathrm{\infty },1}^1(\mathbb{R})$ which leads to ill-posedness.