The Study Of The Blow-up Of Solution Of The Camassa-Holm Equation And The Degasperis-Procesi Equation

Initial boundary value problem of the evolution of nonlinear equations has often been an interesting academic study subject.It includes the existence of solution,the uniqueness of solution,the stability of solution,the blow-up of solution and the regularity of solution.In this paper,we study the existence and the blow-up of solution of the Camassa-Holm Equation,the Degasperis-Procesi equation and the Camassa-Holm Equation with weak dissipation.There are five sections in this paper:The first section,we introduce the background and actuality.The second section,we introduce the basic concept.The third section,we study the critical threshold phenomenon in Camassa-Holm Equation,that is the critical condition for the breakdown of the solutions as well as the global existence of the solutions.Using characteristic curve and the upper and lower solution,we give the upper and lower threshold of the Camassa-Holm equation, the estimator of the blow-up rate is also derived.The fourth section,we study the blow-up problem of the Degasperis-Procesi equation.Under some intial conditions,the solution of the equation occurs blow-up phenomena,and then we investigate the blow-up rate.The fifth section,we study the critical threshold phenomenon in Camassa-Holm Equation with weak dissipation,we give the upper and lower threshold of the Camassa-Holm equation with weak dissipation,and the estimator of the blow-up rate is also derived.