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Transactions of the American Mathematical Society

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Compactness of the solution operator for a linear evolution equation with distributed measures
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by Ioan I. Vrabie PDF
Trans. Amer. Math. Soc. 354 (2002), 3181-3205 Request permission


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
  • Email:
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 3181-3205
  • MSC (2000): Primary 47D06, 46G10, 47B07; Secondary 35A05, 35J99, 35K99
  • DOI:
  • MathSciNet review: 1897396