Study On Blow-up Properties Of Solutions To The Higher-order Camassa-Holm Equation

Blow-up criteria,blow-up rate,blow-up set constitute the solutions of nonlinear partial differential equations,always is one of the fundamental questions of partial differential equations.In this paper,we consider the blow-up properties of solutions to the second-order Camassa-Holm equations.Firstly,by using Holder inequality,Gronwall inequality and a series of prior estimates the blow-up criteria,blow-up rate are gained.Secondly,by using the properties of closed set,the blow-up set is obtained.Finally,the blow-up set is also investigated under some suitable initial date conditions.In first section,we study the blow-up criteria and the precise blow-up rate the second-order Camassa-Holm.In second section,we study the blow-up set of the second-order Camassa-Holm by using the properties on closed set.In third section,we study the no blow-up set of the second-order Camassa-Holm by a series of prior estimates under some suitable initial data which is non-negative or remain the same on R.