Tauberian theorems and stability of solutions

of the Cauchy problem

Authors:
Charles J. K. Batty, Jan van Neerven and Frank Räbiger

Journal:
Trans. Amer. Math. Soc. **350** (1998), 2087-2103

MSC (1991):
Primary 44A10; Secondary 47D06, 47D03

DOI:
https://doi.org/10.1090/S0002-9947-98-01920-5

MathSciNet review:
1422891

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a bounded, strongly measurable function with values in a Banach space , and let be the singular set of the Laplace transform in . Suppose that is countable and uniformly for , as , for each in . It is shown that

as , for each in ; in particular, if is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on , and it implies several results concerning stability of solutions of Cauchy problems.

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Additional Information

**Charles J. K. Batty**

Affiliation:
St. John’s College, Oxford OX1 3JP, England

Email:
charles.batty@sjc.ox.ac.uk

**Jan van Neerven**

Affiliation:
Department of Mathematics, Delft Technical University, P.O. Box 356, 2600 AJ Delft, The Netherlands

Email:
J.vanNeerven@twi.tudelft.nl

**Frank Räbiger**

Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Email:
frra@michelangelo.mathematik.uni-tuebingen.de

DOI:
https://doi.org/10.1090/S0002-9947-98-01920-5

Keywords:
Laplace transform,
Tauberian theorem,
singular set,
countable,
$C_{0}$-semigroup,
stability,
local spectrum,
orbit,
Cauchy problem

Received by editor(s):
February 12, 1996

Received by editor(s) in revised form:
September 6, 1996

Additional Notes:
The work on this paper was done during a two-year stay at the University of Tübingen. Support by an Individual Fellowship from the Human Capital and Mobility Programme of the European Community is gratefully acknowledged. I warmly thank Professor Rainer Nagel and the members of his group for their hospitality (second author). It is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG (third author). Work in Oxford was also supported by an EPSRC Visiting Fellowship Research Grant (first and third authors).

Article copyright:
© Copyright 1998
American Mathematical Society