Date of Award

May 2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Lijing Sun

Committee Members

Dashan Fan, Gabriella Pinter, Lei Wang, Chao Zhu

Keywords

Alpha-Modulation Space, Asymptotic Estimates, Dispersive Equations

Abstract

The alpha-modulation space is a function space developed by Grobner in 1992. The alpha-modulation space is a generalization of the modulation space and Besov space. In this thesis we obtain asymptotic estimates for the Cauchy Problem for dispersive equation, a generalized half Klein-Gordon, and the Klein-Gordon equations. The wave equations will also be considered in this thesis too. These estimates were found by using standard tools from harmonic analysis. Then we use these estimates with a multiplication algebra property of the alpha-modulation space to prove that there are unique solutions locally in time for a nonlinear version of these partial differential equations in the function space of continuous function in time and alpha-modulation in the spatial component. These results are obtained by using the fixed point theorem.

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