Transactions of the American Mathematical Society

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

Author(s): Ioan I. Vrabie
Journal: Trans. Amer. Math. Soc. 354 (2002), 3181-3205.
MSC (2000): Primary 47D06, 46G10, 47B07; Secondary 35A05, 35J99, 35K99
Posted: April 1, 2002
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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

, $u(0)=\xi$ , where

generates a

-semigroup in a Banach space

, $\xi\in X$ , $g\in BV([\,a,b\,];X)$ , 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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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47D06, 46G10, 47B07, 35A05, 35J99, 35K99

Ioan I. Vrabie
Affiliation: Faculty of Mathematics, ``Al. I. Cuza" University of Iasi, Iasi 6600, Romania
Address at time of publication: P. O. Box 180, Ro, Is 1, Iasi 6600, Romania
Email: ivrabie@uaic.ro

DOI: 10.1090/S0002-9947-02-02997-5
PII: S 0002-9947(02)02997-5
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
Posted: 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
Copyright of article: Copyright 2002, American Mathematical Society

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