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In this paper, we study the Cauchy problem for the Camassa-Holm equation onthe real line. By presenting a new construction of initial data, we show thatthe solution map in the smaller space $B_{p,1}^{1}\cap C^{0,1}$ with$p\in(2,\infty]$ is discontinuous at origin. More precisely, $u_0\inB_{p,1}^{1}\cap C^{0,1}$ can guarantee that the Camassa-Holm equation has aunique local solution in $W^{1,p}\cap C^{0,1}$, however, this solution isinstable and can have an inflation in $B_{p,1}^{1}\cap C^{0,1}$ for certaininitial data.

Ill-posedness for the Camassa-Holm equation in $B_{p,1}^{1}\cap C^{0,1}$