Scattering Theory Of Non-local Type Of Dispersive Wave Equation

In this dissertation,we introduce our results on the local and global well-posedness theories and the scattering theory of the dispersive equations in recent years.In the preface,we split nine subsections to introduce the concepts,methods and ideas on the scattering theory for dispersive equations.That is,the concept of dispersive equations,dispersion and Strichartz estimate,symmetry and scaling analysis,the concept of scattering theory,Morawetz estimate,iteration structure for the equation, finite speed of propagation and casuality,and compactness and its characterization.In Chapter 2,we study the decay estimate and scattering theory for the Klein-Gordon -Hartree equations with radial data in space dimension d≥3.By means of a new strategy due to Visciglic and exploiting the linear and nonlinear parts of the Morawetz estimate,we obtain the corresponding theory for energy subcritical and critical cases respectively.The exponent range of the decay estimates are extended to 0＜γ≤4,γ＜d with Hartree potential V(x)=|x|^-γ.In Chapter 3,we study the theory of scattering in the energy space for the Klein-Gordon equation with a cubic convolution in space dimension n≥3.We establish a casuality for nonlocal Klein-Gordon equations which extended the result of Menzala-Strauss [39]and can be viewed as a substitute for the ordinary finite speed of propagation. By means of the strategy of frequency decomposition to distinguish the dispersive effects,the fact that local decay implies local spacetime bound,and the flexibility of the Strichartz estimates for the Klein-Gordon equation,along with the method of Morawetz-Strauss[64]and Ginibre-Velo[28],we prove the asymptotic completeness for some suitable potential assumptions.The results cover in particular the case of the potential |x|^-γ for 2＜γ＜min(4,n).In Chapter 4,We consider the defocusing,energy-critical Hartree equation with harmonic potential for the radial data in all dimensions(n≥5).We show the global well-posedness and scattering results in the space∑=H¹âˆ©FH¹.We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well.Based on the approach of Bourgain-Tao,we use the localized Morawetz identity to show global well-posedness. A key decay estimate comes from the linear part of energy estimate rather than the nonlinear part,which finally gives the complete scattering theory.Chapter 5 studies the real analyticity of the scattering operator for the Hartree equation i(?)_tu=Δu+u(V*|u|²) based on scatter theory in Ginibre-Velo[27].Such results have been known only for the Klein-Gordon equation[1][37].For the Schr(o|¨)dinger equation,exploiting interior and exterior cut-off on time and space and utilizing sufficiently compactness,we overcome the difficulties which arise from absence of good properties for the Klein-Gordon equation,such as the finite speed of propagation and ideal time decay estimate.Additionally,the method simplifies the proof of Theorem 1.1 in Kumlin[37].The method of this paper involves comprehensive analytic techniques.