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Nonuniform stability
of damped contraction semigroups
Ralph Chill, Lassi Paunonen, David Seifert, Reinhard Stahn and Yuri Tomilov |
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Vol. 16 (2023), No. 5, 1089–1132
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Abstract |
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We investigate the stability properties of strongly continuous semigroups generated by operators of the form , where is the generator of a contraction semigroup and 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 in terms of selected observability-type conditions on the pair . 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
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Mathematical Subject Classification 2010
Primary: 47D06, 34D05, 47A10, 35L90
Secondary: 93D15, 35L05
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Milestones
Received: 20 December 2019
Revised: 15 September 2021
Accepted: 19 November 2021
Published: 12 August 2023
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Authors |
| Ralph Chill | |
| Institut für Analysis Fakultät für Mathematik TU Dresden Dresden Germany |
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| Lassi Paunonen | |
| Mathematics Faculty of Information Technology and Communication Sciences Tampere University Tampere Finland |
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| David Seifert | |
| School of Mathematics, Statistics
and Physics Newcastle University Newcastle upon Tyne United Kingdom |
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| Reinhard Stahn | |
| Institut für Analysis Fakultät für Mathematik TU Dresden Dresden Germany |
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| Yuri Tomilov | |
| Institute of Mathematics Polish Academy of Sciences Warsaw Poland |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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