Global Well-posedness For NLS Equation And Fourth Order Nonlinear Klein-Gordon Equation

In Section1, we introduce the physical background and research history of the nonlinearSchro¨dinger equations and Klein-Gordon equations and give an outline of this thesis.In Section2, we consider the Cauchy problem for a class of nonlinear Schro¨dinger equa-tions with critical exponent. The Schro¨dinger equation is the most fundamental equation innon-relativistic quantum mechanics, therefore it is of great value to study this problem. Byconstructing the cross-constrained variational problem and the so-called invariant manifolds,we derive a sharp condition for global existence and nonexistence of solutions. Moreover weanalyze the source term with critical index and supercritical index on global well-posedness ofsolutions to nonlinear Schro¨dinger equations.In Section3, we study the Cauchy problem for nonlinear Schro¨dinger equation with severalnonlinear source terms of different signs. By constructing the cross-constrained variationalproblem and the invariant manifolds of the flow and using the potential well concavity method,we present a sharp condition for global existence and nonexistence of solutions. Furthermoreseveral nonlinear source terms of different signs constitute the complex variational structure,hence it is necessary to consider how the nonlinear source terms influence the space structure.In Section4, we investigate the initial boundary value problem of fourth order nonlin-ear Klein-Gordon equations. On the one hand the existence of the dispersive term makes thepotential structure more complex, so it is very difficult to study the global well-posedness ofsolutions at high energy level. On the other hand there is not only a way to built up the vari-ational structure in the presence of the complex potential structure, hence it is necessary todiscuss how the different variational structures affects the well-posedness problem and analysisthe relations between different variational structures and solutions. By establishing the appro-priate cross-constrain variational problem we present a sharp condition for global existence andnonexistence of solutions at low and critical energy level. Moreover by exploiting the convexmethods, we obtain a blow up result for certain solutions with arbitrary positive energy.