  In this paper, we give a construction of $u_0\in B^\sigma_{p,\infty}$ such
that corresponding solution to the Camassa-Holm equation starting from $u_0$ is
discontinuous at $t = 0$ in the metric of $B^\sigma_{p,\infty}$, which implies
the ill-posedness for this equation in $B^\sigma_{p,\infty}$. We also apply our
method to the b-equation and Novikov equation.
