Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation

In this paper, we mainly study several problems on the weakly dissipative periodic Camassa-Holm equation. At first, the local well-posedness of the equation is obtained by Kato's theorem, a necessary and sufficient condition of the blow-up of the solution and some criteria guaranteeing the blow-up of the solution are established. Then, the blow-up rate of the solution is discussed. Moreover, we prove that the equation has global solutions and these global solutions decay to zero as time goes to infinite provided the potentials associated to their initial date are of one sign.