The Initial Boundary Value Problem Of A Class Of Nonlinear Wave Equations

This dissertation mainly investigates the following properties of solutions of a class of wave equations with a nonlinear source term and a damping term. It contains the problems of existence of solutions and finite time blow up. utt+α△2u-β△u-γ△ut+σut+τu=f(u), (x,t)∈Ω×[0,+∞) u(x,0)=u0(x), ut(x,0)=u1(x), x∈Ω, u(x,t)=0, (x,t)∈(?)Ω×(0,T).WhereΩ(?) Rn is a bounded domain with smooth boundary (?)Ω,u(x, t) is unknown function, △ is the Laplace operator.This thesis consists of four parts:In the first chapter, some physical back ground will be introduced of the nonlinear partial differential equations with initial boundary value problem. Simultaneously, some available results will be surveyed so far.In the second chapter, we study this forth order nonlinear wave equation with the initial boundary value problem and the existence of the global weak solutions to this problem is proved by means of the potential well method and the Galerkin method.The chapter 3 deals with the initial boundary value problem for a class of fourth order damped nonlinear wave equations. By using the potential well method, some results of blow up solutions with certain initial profiles are established.In chapter 4, we investigate the finite time blow up problems for the nonlinear wave equation with the initial boundary value problem when γ=0,f(u)=|u|p-1u at high energy level. By employing the potential well method, this theme gives that for which initial data the solutions of this problem can blow up at high energy level.In the end, we give a conclusion of this paper.