Blow-Up and Decay of the Solution of the Weakly Dissipative Degasperis–Procesi Equation

In this paper, we mainly study several problems on the weakly dissipative Degasperis–Procesi equation. We first establish the local well-posedness of the equation, derive a precise blow-up scenario, and present two blow-up criteria for strong solutions to the equation. We then give the precise blow-up rate of blow-up solutions to the equation. We finally prove that the equation has global strong solutions and these global solutions decay to zero as time goes to infinity provided the potentials associated with their initial data are of one sign.