The Study Of The Initial Boundary Value Problems For Two Classes Of Fourth Order Wave Equations With Damping Terms

This paper studies global well-posedness of solutions for two classes of fourth order wave equations with strong damping and weak damping.In chapter 2,we consider the initial boundary problem of a class of fourth order wave equations with strong damping and nonlinear weak damping.First,we construct some energy functionals and give some preliminary lemmas by using the potential well,and then we obtain the existence and uniqueness of global strong solutions in low initial energy case together with Galerkin method.Furthermore we get the existence,longtime behavior and blow up of the weak solutions when the initial energy is equal to the depth of potential well.In chapter 3,we investigate global existence and blow up of weak solutions for a class of the fourth order wave equations with strong damping,linear damping and viscoelastic term.This chapter obtains the depth of potential well by establishing variational structure of this problem.By using Galerkin method,we construct the approximate solutions of the system and estimate the boundedness of the approximate solutions at low energy case,then we obtain the global existence of weak solutions.Next,we show the weak solutions blow up in finite time by using concavity function method.On this basis,by defining some new conditions and auxiliary functions we get new invariant set of critical initial energy and high initial energy cases,then we extend this conclusion to critical initial energy and high initial energy cases.