Nonlocal Nonlinear Schrodinger Equations In Three Space Dimensions

The nonlocal nonlinear Schro¨dinger equations describe the magneticZakharov system in the infinite ion acoustic speed limit in a cold plasma. Theyhold important physical background and it is valuable to study them mathemat-ically. In this thesis, we study blow-up solution, global solution and standingwave solution for a kind of generalized nonlocal nonlinear Schr¨odinger equationsin three space dimensions. First of all, through establishing proper virial identi-ties and making some a priori estimates on these nonlocal terms, we prove thatthe solution of the Cauchy problem to the nonlocal nonlinear Schr¨odinger equa-tions under consideration blows up in finite time. Next, utilizing the inner struc-ture of the nonlocal nonlinear Schr¨odinger equations under study, overcomingthe difculty that the nonlocal terms destroy the scaling invariance, introducingproper variationala structure and Lagrange multiplier method, constructing suit-able functionals and manifolds, we obtain the existence and orbital instabilityof standing wave for the equations under consideration. Finally, by introducingproper invariant manifolds, we establish the sharp condition of global existencefor the Cauchy problem to the related nonlocal nonlinear Schr¨odinger equations.