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Nonuniform stability of damped contraction semigroups

Ralph Chill, Lassi Paunonen, David Seifert, Reinhard Stahn and Yuri Tomilov

Vol. 16 (2023), No. 5, 1089–1132
Abstract

We investigate the stability properties of strongly continuous semigroups generated by operators of the form A BB , where A is the generator of a contraction semigroup and B is a possibly unbounded operator. Such systems arise naturally in the study of hyperbolic partial differential equations with damping on the boundary or inside the spatial domain. As our main results we present general sufficient conditions for nonuniform stability of the semigroup generated by A BB in terms of selected observability-type conditions on the pair (B,A). The core of our approach consists of deriving resolvent estimates for the generator expressed in terms of these observability properties. We apply the abstract results to obtain rates of energy decay in one-dimensional and two-dimensional wave equations, a damped fractional Klein–Gordon equation and a weakly damped beam equation.

Keywords
nonuniform stability, strongly continuous semigroup, resolvent estimate, hyperbolic equation, observability, damped wave equation, Klein–Gordon equation, beam equation
Mathematical Subject Classification 2010
Primary: 47D06, 34D05, 47A10, 35L90
Secondary: 93D15, 35L05
Milestones
Received: 20 December 2019
Revised: 15 September 2021
Accepted: 19 November 2021
Published: 12 August 2023
Authors
Ralph Chill
Institut für Analysis
Fakultät für Mathematik
TU Dresden
Dresden
Germany
Lassi Paunonen
Mathematics
Faculty of Information Technology and Communication Sciences
Tampere University
Tampere
Finland
David Seifert
School of Mathematics, Statistics and Physics
Newcastle University
Newcastle upon Tyne
United Kingdom
Reinhard Stahn
Institut für Analysis
Fakultät für Mathematik
TU Dresden
Dresden
Germany
Yuri Tomilov
Institute of Mathematics
Polish Academy of Sciences
Warsaw
Poland

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