On the Cauchy problem for the periodic generalized Degasperis–Procesi equation

We mainly study the Cauchy problem of the periodic generalized Degasperis–Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.