Asymptotic Estimates for Some Dispersive Equations on the Alpha-modulation Space

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.