The Study Of Nonexistence Of Peaked Solution And Blow-up Criteria For A Fifth-order Camassa-Holm Equation

As a relatively new completely integrable dispersive shallow-water wave equation,Camassa-Holm equation describes the unidirectional propagation of waves on a stationary shallow water free surface at infinity.Camassa-Holm equation has attracted extensive attention because of its properties of peaked soliton and describing wave breaking phenomenon in the past few decades.In this paper,we mainly study the nonexistence of peaked weak solution and blow-up properties of the strong solution for a fifth-order Camassa-Holm equation.Firstly,we demonstrate the nonexistence of single peaked weak solution for the equation by contradiction.Secondly,by using the transport equation theory and Moser type estimation,a necessary condition for the blow-up of the solution of the equation in finite time is studied through induction.Finally,in two different cases,by estimating the solution,the sufficient conditions for the existence of global solutions of the fifth-order Camassa-Holm equation are studied.