The Study Of Two Class Of Nonlinear Evolution Equations

The initial-boundary value problem for the nonlinear Klein-Gordon equation with competitive term is studied in this paper.By introducing certain potential well,some sufficient conditions for the global existence and blow-up results to the solution are given.Furthermore,the upper bound of life span is also obtained while the solution blows up.The global well-posedness of a class of fourth order nonlinear Schr(?)dinger equations is also considered.One of the novel aspects of this work is combined that the application of a new interpolation-embeding inequality which now is know as B-G inequality and Galerkin methods.The arrangement of this paper is as follows:In the first chapter,some backgrounds,known methods and the main results of this paper are given.In the second chapter,some basic definitions about potential well,well depth and some lemmas are given.Then,the conditions of global existence and blow-up of solution of nonlinear Klein-Gordon equation when the initial energy is equal to or less than potential well are proved by using the Galerkin method,and the estimations of the upper bound of life span are given when the solution blows up.The conservation of mass and energy are given in the third chapter.With the help of the B-G inequality and Gagliardo-Nirenberb inequality,the global existence and uniqueness of solution of nonlinear Schr(?)dinger equations are obtained by using Galerkin method.