Time-space Estimate Method For Several Kinds Of Nonlinear Evolution Equation(s)

In this thesis, we use time-space estimate method to deal with several kinds of nonlinear evolution equation(s). It's well-known that once L~p - L~q(L~p- L~p') estimate is constructed on the basis of oscillation integral, we can obtain time-space estimate by TT~* method for linear equation. Firstly, we recall p - p' estimates and time-space estimates of three classical equations (Schrodinger equation, wave equation and Klein-Gordon equation), which are necessary tools to continue our research.Klein-Gorden-schrodinger equationsdescribe a classical model of Yukawa's interaction of conversed complex nucleon field with neutral real meson field. Here u represents a complex scalar nucleon field and v a real scalar meson field. The real constant μ describes the mass of a meson.For well-posedness of (0.1), one may refer readers to [2, 23, 32, 63]. In 1985, J. Ginibre & G. Velo considered the well-posedness in energy-space of pure Klein-Gordon and pure Schrodinger equations in their papers [27, 28]. Through imposing assumptions on nonlinearity f, they obtained the global existence of weak solutions by Galerkin method. Under stronger assumptions, they obtain the uniqueness of those solutions by usage of Strichartz estimates and estimates for nonlinearity. Inspired by this approach, we beginning to consider K-G-S equations. At the same time, we are confronted with several questions. How to generalize the nonlinearities of (0.1)? How to impose assumptions on nonlinearities to ensure the global existence of weak solutions? The method depends on Strichartz estimate while the admissible pairs of two kinds of equations are different, how to choose the admissible pairs? Can we remove the unnatural restriction on the growth exponent of nonlinearities? All of these questions are to be answered one by one with our further research.In the first part of this thesis, we consider well-posedness of generalized nonlinear K-G-S...