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\begin{document}

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\begin{center}
{\bf Stability estimates for semigroups and partly parabolic reaction diffusion equations}\vspace{20pt}

\rule{4in}{1pt}\vspace{20pt}

{\bf A Dissertation

 presented to

 the Faculty of the Graduate School

 University of Missouri}\vspace{20pt}

\rule{4in}{1pt}\vspace{20pt}

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy \vspace{20pt}

\rule{4in}{1pt}\vspace{20pt}

by

VALERIAN YUROV

Dr. Yuri Latushkin, Dissertation Supervisor\vspace{20pt}

JULY 2013\end{center}\newpage


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The undersigned, appointed by the Dean of the Graduate School, have examined
the dissertation entitled

\begin{center}Stability estimates for strongly continuous semigroups and partly parabolic reaction diffusion equations \end{center}

presented by Valerian Yurov, a candidate for the degree of Doctor of Philosophy of

Mathematics, and hereby certify that in their opinion it is worthy of acceptance.

\hspace{1.875in}\parbox{4in}{\begin{center}

\vspace{42pt}\rule{4in}{1pt}\vspace{.125in}\\Professor Yuri Latushkin

\vspace{42pt}\rule{4in}{1pt}\vspace{.125in}\\Professor Carmen Chicone

\vspace{42pt}\rule{4in}{1pt}\vspace{.125in}\\Professor Fritz Gesztesy

\vspace{42pt}\rule{4in}{1pt}\vspace{.125in}\\Professor Michael Pang

\vspace{42pt}\rule{4in}{1pt}\\Professor David Retzloff \end{center}}
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\begin{center}ACKNOWLEDGEMENTS\end{center}
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First of all, I wish to express sincere gratitude to my advisor, Professor Yuri Latushkin, for his vision and direction. His priceless gift to me was not just skills and knowledge, but also an unwavering moral support through every stage of my doctoral studies. His teachings and encouragement will continue to instruct and inspire long beyond my years as a graduate student.

I would also like to thank my committee members Dr. Carmen Chicone, Dr. Fritz Gesztesy, Dr. Michael Pang and Dr. David Retzloff  for their support and encouragement.

I sincerely thank Dr.Anna Gazaryan and Dr.Jens Rottman-Matthes whose excellent articles has motivated and helped current project in more ways than one. A very special thanks are also reserved for Sergey Tikhomirov for his crucial suggestions in the proof of certain inequality in the last chapter of this thesis.

Finally, I am very grateful to MU Mathematics department's academic support staff Amy Crews and Gwen Gilpin. Their timely help and advices are deeply appreciated.

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\begin{center}Stability estimates for strongly continuous semigroups and partly parabolic reaction diffusion equations

Valerian Yurov

Dr. Yuri Latushkin, Dissertation Supervisor

ABSTRACT\end{center}%\addcontentsline{toc}{chapter}{Abstract}

The purpose of this thesis is to apply methods of the abstract theory of strongly continuous operator semigroups and of evolution semigroups to study the spectral properties of a class of differential operators on the line that appears when one linearizes partial differential equations about such special solutions as steady states or traveling waves.

We begin by discussing the stability of the traveling wave solutions of a reaction-diffusion system with a degenerate diffusion matrix. We demonstrate that under some reasonable assumptions on the system, its spectral stability directly implies the linear stability. In particular, we study asymptotic spectral properties of certain first order matrix differential operators, thus generalizing some results known for the evolution semigroups.

We then turn to abstract strongly continuous operator semigroup on Banach spaces, revisit a quantitative version of Datko's Stability Theorem and obtain the estimates for the constant $M$ satisfying the inequality $\|T(t)\|\le M\eb^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution, $L^p$-Fourier multipliers, and other operators involved in Datko's Stability Theorem. This generalizes recent results for the Hilbert spaces on estimating $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.

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\chapter{Introduction}  \lb{ch1}

\chapter{A spectral mapping result for partly parabolic systems} \lb{ch2}

\section{Preliminaries}\label{sec2.1}

\subsection{Semigroups}\label{subs2.1.1}



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\begin{thebibliography}{xxxxxx}
\normalsize

\bibitem[A]{A} H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, {\em Math. Nachr.}, {\bf 186} (1997), 5 -- 56.

\end{thebibliography}

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