Global Existence And Blow-up Of Solutions For Cauchy Problem Of Nonlinear Klein-Gordon Equations

In this paper,we study the Cauchy problem of nonlinear Klein-GordonequationsBy introducing a family of potential wells, we not only give a threshold resultof global existence and nonexistence of solutions, but also obtain the vacuumisolating of solutions. Finally we prove the global existence of solutions for aboveproblem with critical initial condition E(0)=d.This thesis mainly consists of five parts:In section one, there is an outline and an introduction of this work. It simplyintroduces the background of partial differential equations and the history ofnonlinear evolution equations. And the main results that we have known on theproblem are given.In section two, we introduce the potential well and its properties.In this part,we first give the concept of potential well and the definition of the family ofpotential wells.Then we give and prove some theorems and lemmas on potentialwell.The third part is about the vacuum isolating of solutions and the invariablesets of solutions.And the vacuum isolating was first shown by Liu Yacheng in2003.In section four, we study the existence of the global weak solutions andblow-up of question (1)—(3).In section five, we study the existence of the global weak solutions andblow-up of question (1)—(3) with critical initial condition E(0)=d.