Convolution inequalities and applications to partial differential equations.

Abstract

In this dissertation we develop methods for obtaining the existence of mild solutions to certain partial differential equations with initial data in weighted L p spaces and apply them to some examples as well as improve the solutions to some known PDEs studied extensively in the literature. We begin by obtaining a version of a Stein-Weiss integral inequality which we will use to obtain general convolution inequalities in weighted L p spaces using the techniques of interpolation. We will then use these convolution inequalities to make estimates on PDEs that will help us obtain mild solutions as fixed points of certain contraction mappings. Then Lorentz spaces will be introduced and interpolation will be used again to obtain convolution inequalities in weighted Lorentz spaces. Finally, the possibility of investigating PDEs with initial data in weighted Lorentz spaces will be discussed.