Global Well-posedness For Wave Equations, Heat Equations And Nonlinear Schr(?)dinger Equations

This paper studies with three classes of problems: the initial boundary value problem for a class of hyperbolic equations with some nonlinear generalized source terms, the initial boundary value problem for a class of parabolic equations with some nonlinear generalized source terms and the initial value problem for a class of nonlinear Schr(?)dinger equations with generalized harmonic potential and nonlinear double source terms of different signs. The nonlinear general-ized source terms of this thesis not only contain a single nonlinear source term and the nonlinear double source terms of same signs, but also have a class of nonlinear double source terms of different signs. This paper discusses the global existence and finite time blow up of solutions for the hyperbolic equations and parabolic equations as well as the Schr(?)dinger equations at the low initial energy level, respectively.For the initial boundary value problems of solutions for the hyperbolic equations and parabolic equations with nonlinear generalized source terms, this paper introduce a single poten-tial well and outer space of potential well and analyze the structural framework of this potential well. From the analysis of the physical model of the problem, we derive some properties of the energy functionals and the Nehari functionals and obtain the invariant set under the flow of this problem. In the end, by using Galerkin method and convex method, we establish some sufficient conditions on the initial data such that the weak solutions exist globally or blow up in finite time at the low initial energy level respectively and analyze the effect of the nonlinear generalized source terms on the global well-posedness of solutions at the low initial energy level.In the end,the global well-posedness problem of solutions for a class of nonlinear Schr(?)dinger equations with complex nonlinear combined power source terms and generalized harmonic potential is considered. The nonlinear Schr(?)dinger equations with complex nonlin-ear combined power source terms and generalized harmonic potential have a wide range of application, which can be derived from fluid mechanics and plasma physics and describe Kerr optical beam propagation. Complex nonlinear combined power source terms and generalized harmonic potential make the behavior of solutions (such as the invariant sets) change, so the standard research for the well-posedness is not valid. In order to resolve difficulty on the well-posedness of solutions from the existence of complex nonlinear combined power source terms and generalized harmonic potential, we redefine the functionals space of solutions and analyze the properties of Nehari functionals and the invariant set of solutions carefully. By using the energy estimate the global existence of solutions is proved and a finite time blow up result is obtained together with a family of potential wells and convex method.