Improved blow-up criteria for some Camassa-Holm type equations

We study the blow-up phenomena for some integrable Camassa-Holm type equations on the line. For the two-component Camassa-Holm system, we give a sufficient condition on the initial data that leads to a blow-up. For the Degasperis-Procesi equation, we establish a local-in-space blow-up criterion which improves considerably the early criterion based on the sign-changing momentum. Besides, we obtain some new blow-up criteria for the Novikov equation and the modified Camassa-Holm equation.