It was proved in [26, J. Differ. Equ., 390 (2024)] that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on initial data in Besov spaces \(B_{p, r}^s({\mathbb {R}})\) with \(s>1\) and \(1\le p,r<\infty \) , which left an open problem in the end-point case \(p=\infty \) . In this paper, we show that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on initial data in \(B_{\infty , r}^s({\mathbb {R}})\) with \(s>1\) and \(1\le r<\infty \) .
