Nonuniform stability of damped contraction semigroups

Ralph Chill; Lassi Paunonen; David Seifert; Reinhard Stahn; Yuri Tomilov

doi:10.2140/apde.2023.16.1089

Nonuniform stability of damped contraction semigroups

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

Computing Sciences

Research output: Contribution to journal › Article › Scientific › peer-review

14 Citations (Scopus)

30 Downloads (Pure)

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.

Original language	English
Pages (from-to)	1089-1132
Number of pages	44
Journal	Analysis and PDE
Volume	16
Issue number	5
DOIs	https://doi.org/10.2140/apde.2023.16.1089
Publication status	Published - 2023
Publication type	A1 Journal article-refereed

Funding

The research of Paunonen is funded by the Academy of Finland grants 298182 and 310489. The work of Tomilov was partially supported the NCN grant UMO-2017/27/B/ST1/00078. MSC2010: primary 47D06, 34D05, 47A10, 35L90; secondary 93D15, 35L05. Keywords: nonuniform stability, strongly continuous semigroup, resolvent estimate, hyperbolic equation, observability, damped wave equation, Klein–Gordon equation, beam equation.

Funders	Funder number
Academy of Finland	310489, 298182
Narodowe Centrum Nauki	UMO-2017/27/B/ST1/00078

Keywords

beam equation
damped wave equation
hyperbolic equation
Klein–Gordon equation
nonuniform stability
observability
resolvent estimate
strongly continuous semigroup

Publication forum classification

Publication forum level 3

ASJC Scopus subject areas

Analysis
Numerical Analysis
Applied Mathematics

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10.2140/apde.2023.16.1089Licence: CC BY

apde-v16-n5-p01-sFinal published version, 901 KBLicence: CC BY

https://urn.fi/URN:NBN:fi:tuni-2023121510879Licence: CC BY

Cite this

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title = "Nonuniform stability of damped contraction semigroups",

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{\textendash}Gordon equation and a weakly damped beam equation.",

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

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T1 - Nonuniform stability of damped contraction semigroups

AU - Chill, Ralph

AU - Paunonen, Lassi

AU - Seifert, David

AU - Stahn, Reinhard

AU - Tomilov, Yuri

N1 - Publisher Copyright: © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - beam equation

KW - damped wave equation

KW - hyperbolic equation

KW - Klein–Gordon equation

KW - nonuniform stability

KW - observability

KW - resolvent estimate

KW - strongly continuous semigroup

U2 - 10.2140/apde.2023.16.1089

DO - 10.2140/apde.2023.16.1089

M3 - Article

AN - SCOPUS:85170202531

SN - 2157-5045

VL - 16

SP - 1089

EP - 1132

JO - Analysis and PDE

JF - Analysis and PDE

IS - 5

ER -

Nonuniform stability of damped contraction semigroups

Abstract

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Keywords

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