  In this paper we consider the $b$-family of equations on the torus $u\_t-
u\_{txx}+ (b+1) u u\_x=b u\_x u\_{xx} + u u\_{xxx}$, which for appropriate
values of $b$ reduces to well-known models, such as the Camassa-Holm equation
or the Degasperis-Procesi equation. We establish a local-in-space blow-up
criterion
