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We prove norm inflation and hence ill-posedness for a class of shallow waterwave equations, such as the Camassa-Holm equation, Degasperis-Procesi equationand Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even inthe Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. Ourresults cover both real-line and torus cases (only real-line case for Novikov),solving an open problem left in the previous works (\cite{Danchin2,Byers,HHK}).

Ill-posedness of the Camassa-Holm and related equations in the critical space