| The dissertation is devoted to studying the global well-posedness, blow up and scattering theory of two kinds of dispersive equations with the non-local nonlinearity by making use of some modern analysis, such as Littlewood-Paley theory, the variational method, the profile-decomposition and so on. The study-ing of scattering theory originates from Segal’s conjecture [84]. Since the 1970s, scattering theory for the nonlinear dispersive equations has become a challeng-ing topic in partial differential equations and physics, see Cazenave’s and Tao’s monograph. From a physical point of view, research on scattering theory is a very useful method for scientists to probe the microscopic nature. And it plays an important role in the researches on quantum physics, chemistry and biology. Prom a mathematical point of view, scattering theory is to study the long time behavior of the solution of nonlinear dispersive equations.Chapter 2 is preliminaries, including some notations, definitions as well as some basic harmonic tools.In Chapter 3, we present a comprehensive study of the Cauchy problem for the generalized Davey-Stewartson system where Under certain conditions with λ1 and λ2, we provide a complete picture of local and global well-posedness, scattering, blow-up of the solution in the energy space. The methods used in the chapter are based upon the perturbation theory from T. Tao, M. Visan and X. Zhang [86] and the convexity method of Glassey [36].In Chapter 4, we study the scattering theory of the solution for the modified Davey-Stewartson system where The main difficulties are the lack of scaling invariance, the failure of the interaction Morawetz estimate and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). To over this difficulties, we utilize the strategy derived from concentration-compactness idea, which was firstly introduced by Kenig-Merle [43].In Chapter 5, we study the scattering theory of the solution for the gener-alized Davey-Stewartson system where The main difficulties are the failure of the interaction Morawetz estimate and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). To over this difficulties, we still utilize the strategy derived from the concentration-compactness method [43].In Chapter 6, we consider the nonlinear Schrodinger equation with the cou-pled effect in the energy space H1(R3). We firstly use a variational approach to give a dichotomy of scattering and blow-up for the radial solution with the energy below the threshold, which is given by the ground state W for the energy-critical NLS: iut+△u=-|u|4u. The basic strategy is still the concentration-compactness arguments from Kenig and Merle [43]. We overcome the main difficulties coming from the lack of scaling invariance and the non-local property of the convolution term. Our result shows that the focusing,H1-critical term-|u|4u plays the decisive role for the threshold of the scattering solution in the energy space. |
