The Nonlinear Dispersion Equation, Wave Equation Self-similar Solutions With Low Regularity

In this dissertation, we use the tool of harmonic analysis to study the problem of self-similar solution and low regularity of wave and dispersive wave equations.In Chapter 2, Chapter 3, we establish the generalized well-posedness of the Cauchy problem of complex Ginzberg-Landau(CGL) equation, which includes the self-similar solutions. Furthermore, we also prove the self-similar solutions to a group of CGL equations converge to the self-similar solution to the limiting equation under suitable conditions.In Chapter 4, we give the new nonlinear estimates in spaces of Besov type. Using the estimates, we prove the global well-posedness of Schr dinger equation with general nonlinear term. This extends the known well-poseness results.In Chapter 5, we study the self-similar solution of nonlinear wave equation. By using the nonlinear estimate in Chapter 4 together with the Strichartz estimates of linear wave equation, we prove the global well-posedness of the self-similar solution of the wave equation with general nonlinear term.In Chapter 6, the Cauchy problem of Schr(?)dinger equation with the initial data possessing infinite L2 norm is considered. Using the method introduced by Bourgain, we prove the Cauchy problem of Schr(?)dinger equation is global well-posed for a class of initial data as the above. Since the global well-posedness of Schr(?)dinger equation in Hs, s < 0 has never been obtained to our best knowledge, our result is an improvement in some sense.In Chapter 7, we consider the global well-posedness of two dimensional Schr(?)dinger-Boussinesq coupled system when the initial data belong to Hs, 0 < s < 1. We show the solution is globally wellposed when s > 8/(11), and is locally wellposed when s > -1/4.