| This dissertation is devoted to the study of the local well-posedness and long-time dynamics of the solutions to dispersive equations.In the study fields of modern partial differential equations and physics,nonlinear dispersion equations are typical mathematical and physical models,of which the solution dynamics is extremely rich,see[71,92,106,131].In Chapter 1,we introduce the study background of nonlinear Schrodinger equations and nonlinear wave equations and the main results.We also present some basic knowledge.In Chapter 2,we consider the well-posedness of the solutions to the non-linear Schrodinger equations on inhomogeneous mediums.By establishing the nonlinear estimates in the critical spaces on Zoll manifolds,we prove the local well-posedness of the solutions to the nonlinear Schrodinger equations in the critical Sobolev spaces.In Chapter 3,we consider the nonlinear wave equations on Eucildean spaces,and prove the global well-posedness and scattering theory for the radial solution with initial data in the critical Besov spaces but without uniform boundedness on the critical Sobolev norm given by the conservation quantities or an a priori assumption in five dimension.Our proof is based on exploiting the structure for radial solutions to the linear wave equation,developing some new Strichartz estimates with respect to Besov spaces and incorporation of the hyperbolic co-ordinate transform and the corresponding Morawetz-type estimates.In Chapter 4,we consider the longtime dynamics of the solutions to the focusing energy-critical Schrodinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in dimension 4.By the variational methods,we divide the regions under the energy threshold into two subregions:by the convex argument,we can find the existence of finite time blowup solutions in one of the subregions;by making use of the concentration compactness methods developed by[72,73,104,105]and interaction Morawetz estimates in[30,36],we prove the global existence and scattering theory for the solutions in the other subregion. |
