The Study Of Solution To The Generalized Hyperelastic-rod Equation

In this paper,we study the Cauchy problem of the existence of the local solution,the Blow-up of solution for the generalized hyperelastic -rod equation,and the characters of it's solution to initial boundary problem,There are three sections in this paper:The first section,we will introduce the background and actuality and summerize the main results.The second section,we consider the Cauchy problem of the generali -zed hyperelastic-rod equation,with Kato's method for abstract quasilinear evolution equations and a prior estimates,we get the existence of the local solution,the Blow-up of solution and it's exact blow-up rate.Finally,we consider the initial boundary problem of the generalized hyperelastic-rod equation,the existence of global solution,H~1 -global exponential stability and H~2 -global asymptotic stability are obtained in this section.