Blow-up to a shallow water wave model including the Degasperis-Procesi equation

A nonlinear equation, depicting motions of shallow water waves and including the famous Degasperis-Procesi model, is considered. The key element is that we derive L2 conservation law of solutions for the nonlinear equation, which leads to the bound of the solution itself. Using several estimates derived from the model, we obtain that when its solution blows up in the Sobolev space if and only if the space derivative of the solution tends to minus infinite.