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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 1422891
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Abstract: Let $f : \mathbb{R}_{+} \to X$ be a bounded, strongly measurable function with values in a Banach space $X$, and let $iE$ be the singular set of the Laplace transform $\widetilde f$ in $i\mathbb{R}$. Suppose that $E$ is countable and $\alpha \left \| \int _{0}^{\infty }e^{-(\alpha + i\eta ) u} f(s+u) \, du \right \|\break \to 0$ uniformly for $s\ge 0$, as $\alpha \searrow 0$, for each $\eta $ in $E$. It is shown that

\begin{displaymath}\left \| \int _{0}^{t} e^{-i\mu u} f(u) \, du - \widetilde f(i\mu ) \right \| \to 0\end{displaymath}

as $t\to \infty $, for each $\mu $ in $\mathbb{R} \setminus E$; in particular, $\|f(t)\| \to 0$ if $f$ 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 $BUC(\mathbb{R}_{+}, X)$, 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

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

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

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