On Global Existence, Blow Up And Life Span To Some Higher Order Nonlinear Hyperbolic Equation

In this paper, we consider the following Cauchy problem of four order nonlinear hyperbolic equationThe global existence and nonexistence for the Cauchy problem with the initial data having critical energy. Meantime, we obtain the super bounds of life span to the Cauchy problem problem for initial energy less than critical value.In chapter 1, the backgroud and the previous results to the Cauchy problem of this kind of equations are introduced. By the way, some results and the method used in this paper also given.In chapter 2, we prove the existence and uniqueness of local solution, by Galerkin method, and deduce that continuity for certain norm of the solution with time variable.In chaper 3, we discusses the equilibrium solution with the critical initial energy value by using the extreme principle of nonlinear functional. Furthermore, we gives the existence and nonexistence of global solutions with the initial energy of critical value with the Sobolev embedding theorem, convexity analysis and energy estimation.In chaper 4, we discuss blow up of solution for initial energy less than critical value by using convexity annlysis, and obtain the super bounds of life span with the method of some differential inequality.