Date on Master's Thesis/Doctoral Dissertation

1-2020

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Swanson, David

Committee Co-Chair (if applicable)

Brown, David

Committee Member

Brown, David

Committee Member

Hu, Changbing

Committee Member

Larson, Lee

Author's Keywords

partial differential equations; convolution inequalities; interpolation; functional analysis

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.

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