Study On The Solutions For A Dissipative Camassa-Holm Equation

In this paper, we study the Degasperis-Procesi equation with dissipative term λ(ux uxx). The local well-posedness is obtained by the Kato’s theorem and the blow-up mech-anism is proved. Two results of blow-up solutions with certain initial profles are estab-lished. The blow-up rate of the blow-up solutions is studied.Persistence properties isstudied fnally. The dissertation is divided into four chapters.In the frst chapter,we introduce the research background of the dissipative Degasperis-Procesi equation and the main results of the paper.In the second chapter,we establish the locally well-posedness for the Degasperis-Procesi equation with dissipative term λ(ux uxx) by Kato’s theorem.In the third chapter,we prove the blow-up mechanism of the equation and give onesufcient conditions that lead to the blow up of the solutions.We show fnally that theblow-up rate of the blow-uping solutions is-1.In the forth chapter, We study the persistence properties of the solution of the equa-tion and reveal the relations between the decay of solution and the one of initial valueu0(x).