On Potential Wells And Applications To Two Kinds Of Nonlinear Wave Equations

In this paper,we study the initial boundary value problem of nonlinear beam equations and the wave equations with two nonlinear source terms of different signsandBy using a new method, we introduce a family of potential wells which include the well-known potential well as a special case, not only give a threshold result of global unique existence of solutions of problem (0.1)and (0.2), but also obtain the behavior of vacuum isolating and blow-up of solutions of problem (0.1). Then the known results are improved very much. Our main work is stated as follows:In chapter 1, firstly, the background, current advancement and research methods for two classes of wave equations are introduced respectively. Then some important results concerned are presented.In chapter 2, firstly, the properties of the family of potential wells are established. Then, using these properties and the Galerkin method we prove the theorem of globe existence of solutions of problem (0.1). Furthermore, the uniqueness of the globe solutions is proved by adopting the Lagrange mean value theorem and the Gronwall inequality. Chapter 3 is concerned with the behavior of vacuum isolating of solutions of problem (0.1) by use of the properties of the family of potential wells.In chapter 4, by constructing unstable set and use the convexity method, the properties of blow-up of the globe solutions of problem (0.1) are investigated.In chapter 5, the existence and uniqueness of globe solutions of problem (0.2) are proposed by adopting the same method as that in chapter 2.