Existence And Uniqueness Of Global Solutions For Fractional Klein-Gordon-Schr(?)dinger Equations

The Schr???dinger equation is the fundamental equation of physics describing nonrelativistic quantum mechanical behavior,which was formulated by the Austrian physicist Erwin Schr???dinger in 1925.In this paper,we study two types of the nonlinear fractional Klein-Gordon-Schr???dinger equation with initial periodical boundary value conditions.By using the energy estimate method,we obtain the existence and uniqueness of weak solution to the above two systems.This dissertation is composed of the following four chapters:In chapter 1,we introduce the background knowledge of the partial differential equations and give the physical background and research trends for the fractional Schr???dinger equation.Besides,we recall some required notations,definitions,lemmas and present the structure of this dissertation.In chapter 2,we consider the nonlinear fractional Klein-Gordon-Schr???dinger equation in three dimensional space,with a priori estimates of the solutionyin H^a?7?W?8?.We prove the existence and uniqueness of the global solution of this problem by using the Gelerkin approximations.In chapter 3,we discuss the nonlinear fractional Klein-Gordon-Schr???dinger equation with initial periodic boundary conditions.We establish a priority estimates for the weak solution?y,f,f_t?for the equation in the space ofH^a?W??H^a?W??L²?W?,H^2a?W??H^2a?W??H^a?W?,H^3a?W??H^3a?W??H^2a?W?respectively.Meanwhile,according to Galerkin method,we prove the existence and uniqueness of weak solution in the space ofH^3a?W??H^3a?W??H^2a?W?.In chapter 4,we summarize the work we have done in this dissertation,and point out the future research directions.In this dissertation,the key point is to get uniform a priori estimates for time about the fractional Klein-Gordon-Schr???dinger equations.As dealing with the estimates,we meet some problems which are difficult to solve by using the standard method.Yet,we overcome them by establishing the dedicate estimates.