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
Full-text PDF
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