Table of Contents
International Journal of Partial Differential Equations
Volume 2013, Article ID 207581, 5 pages
http://dx.doi.org/10.1155/2013/207581
Research Article

Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

Received 18 February 2013; Revised 5 April 2013; Accepted 16 April 2013

Academic Editor: Pavel A. Krutitskii

Copyright © 2013 Michael Gil'. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a generator of an exponentially stable operator semigroup in a Banach space, and let be a linear bounded variable operator. Assuming that is sufficiently small in a certain sense for the equation , we derive exponential stability conditions. Besides, we do not require that for each , the “frozen” autonomous equation is stable. In particular, we consider evolution equations with periodic operator coefficients. These results are applied to partial differential equations.