On the finite time singularities for a class of Degasperis-Procesi equations

As we all know, the conservation laws take an important part in deriving wave breaking for Degasperis-Procesi (DP) type equation. In the article, we give a new method to study singularities in finite time for a class of DP equations with high nonlinear terms. The paper tells us that the result of wave breaking can be established without the conservation law.