The Cauchy problem of the modified CH and DP equations

We mainly study the Cauchy problem of modified Camassa-Holm and Degasperis-Procesi equations. First, we establish the local well-posedness for the equation in Besov space. Secondly, we derive the conservation laws and a precise blow-up scenario. Moreover, we prove the existence of blow-up solutions and obtain its blow-up rate, provided the initial data satisfy certain conditions. Finally, we present the persistence properties of the equation.