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Compactness of the solution operator for a linear evolution equation with distributed measures

Author: Ioan I. Vrabie
Journal: Trans. Amer. Math. Soc. 354 (2002), 3181-3205
MSC (2000): Primary 47D06, 46G10, 47B07; Secondary 35A05, 35J99, 35K99
Published electronically: April 1, 2002
MathSciNet review: 1897396
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Abstract: The main goal of the present paper is to define the solution operator $(\xi,g)\mapsto u$associated to the evolution equation $du=(Au)dt+dg$, $u(0)=\xi$, where $A$generates a $C_0$-semigroup in a Banach space $X$, $\xi\in X$, $g\in BV([\,a,b\,];X)$, and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution operator extending some earlier results due to the author and to BARAS, HASSAN, VERON, as well as some applications to the existence of certain generalized solutions to a semilinear equation involving distributed, or even spatial, measures, are also included. Two concrete examples of elliptic and parabolic partial differential equations subjected to impulsive dynamic conditions on the boundary illustrate the effectiveness of the abstract results.

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Additional Information

Ioan I. Vrabie
Affiliation: Faculty of Mathematics, “Al. I. Cuza" University of Iaşi, Iaşi 6600, Romania
Address at time of publication: P. O. Box 180, Ro, Iş 1, Iaşi 6600, Romania

Keywords: Linear evolution equation, $C_0$-semigroup, vector-valued function of bounded variation, compactness of the solution operator
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: September 21, 2001
Published electronically: April 1, 2002
Additional Notes: This research was supported in part by the CNCSU/CNFIS Grant C120(1998) of the World Bank and the Romanian Government
Article copyright: © Copyright 2002 American Mathematical Society

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