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For the famous Camassa-Holm equation, the well-posedness in$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty)$ and theill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $ p\in [1,\infty],\r\in (1,\infty]$ had been studied in \cite{d1,d2,glmy,yyg}, that is to say, itonly left an open problem in the critical case $B^{1}_{\infty,1}(\mathbb{R})$proposed by Danchin in \cite{d1,d2}. In this paper, we solve this problem byproving the norm inflation and hence the ill-posedness for the Camassa-Holmequation in $B^{1}_{\infty,1}(\mathbb{R})$. Therefore, the well-posedness andill-posedness for the Camassa-Holm equation in all critial Besov spaces$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty]$ have beencompleted. Finally, since the norm inflation occurs by choosing an specialinitial data $u_0\in B^{1}_{\infty,1}(\mathbb{R})$ but $u^2_{0x}\notinB^{0}_{\infty,1}(\mathbb{R})$ (an example implies$B^{0}_{\infty,1}(\mathbb{R})$ is not a Banach algebra), we then prove thatthis condition is necessary. That is, if $u^2_{0x}\inB^{0}_{\infty,1}(\mathbb{R})$ holds, then the Camassa-Holm equation has aunique solution $u(t,x)\in \mathcal{C}_T(B^{1}_{\infty,1}(\mathbb{R}))\cap\mathcal{C}^{1}_T(B^{0}_{\infty,1}(\mathbb{R}))$ and the norm inflation willnot occur.

Ill-posedness for the Cauchy problem of the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$