Global Existence Of Two Classes Of Nonlinear Schr?dinger Equations

In this paper,we study the global existence of two classes of non-autonomous nonlinear Schr?dinger equations.The non-autonomy of equations leads to some essential mathematical difficulties.For example,the energy is no longer conserved,Scaling invariance and Galilean,invariance do not hold,and so on.This thesis is divided into four sections according to contents.In Chapter 1,we introduce some background,methods and well-known results of classical and non-autonomous nonlinear Schr?dinger equations.We also give a comparison between the classical and non-autonomous equations.In Chapter 2,we present some preliminaries.In Chapter 3,by using the method of energy estimate,we firstly establish some sufficient conditions about global existence for the non-autonomous equation iut =-1/2?u+V(x)u + f(t)|u|2?u-ig(t)|u|2pu.and then obtain the some sufficient condi-tions about global existence for the Gross-Pitaevskii equation with linear pumping and nonlinear damping which reads i?t=-1/2??+V(x),?+?|?|2??+i(a-b|?|2p)?.In Chapter 4,we study the global existence for the fractional Hartree equation with time-dependent loss/gain.We discuss the global existence of this equation from two aspects,the loss/gain coefficients and the initial data,respectively.By using a bootstrap argument,we obtain a global existence result which depends on the size of the loss/gain coefficients.We also get the other global existence result depending on the initial data.These results extend and improve some earlier results.