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

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