The damped wave equation with singular damping

Pedro Freitas; Nicolas Hefti; Petr Siegl

doi:10.1090/proc/15063

The damped wave equation with singular damping

Pedro Freitas, Nicolas Hefti, Petr Siegl

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

Abstract

We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form α/x, α > 0. We establish the exponential stability of the semigroup for all positive α, and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.

Original language	English
Pages (from-to)	4273-4284
Number of pages	12
Journal	Proceedings of the American Mathematical Society
Volume	148
Issue number	10
DOIs	https://doi.org/10.1090/proc/15063
Publication status	Published - Oct 2020
Externally published	Yes

Keywords

damped wave equation
singular damping
empty spectrum
finite-time extinction
Laguerre polynomials

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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title = "The damped wave equation with singular damping",

abstract = "We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form α/x, α > 0. We establish the exponential stability of the semigroup for all positive α, and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.",

keywords = "damped wave equation, singular damping, empty spectrum, finite-time extinction, Laguerre polynomials",

author = "Pedro Freitas and Nicolas Hefti and Petr Siegl",

note = "Funding Information: Received by the editors June 28, 2019, and, in revised form, February 9, 2020. 2010 Mathematics Subject Classification. Primary 35P15, 35L05. Key words and phrases. Damped wave equation, singular damping, empty spectrum, finite-time extinction, Laguerre polynomials. The first and third authors were partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/2014. The third author was also partially supported by the Swiss National Science Foundation Ambizione grant No. PZ00P2 154786 (until December 2017). Publisher Copyright: {\textcopyright} 2020 American Mathematical Society",

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month = oct,

doi = "10.1090/proc/15063",

language = "English",

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pages = "4273--4284",

journal = "Proceedings of the American Mathematical Society",

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AU - Freitas, Pedro

AU - Hefti, Nicolas

AU - Siegl, Petr

N1 - Funding Information: Received by the editors June 28, 2019, and, in revised form, February 9, 2020. 2010 Mathematics Subject Classification. Primary 35P15, 35L05. Key words and phrases. Damped wave equation, singular damping, empty spectrum, finite-time extinction, Laguerre polynomials. The first and third authors were partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/2014. The third author was also partially supported by the Swiss National Science Foundation Ambizione grant No. PZ00P2 154786 (until December 2017). Publisher Copyright: © 2020 American Mathematical Society

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N2 - We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form α/x, α > 0. We establish the exponential stability of the semigroup for all positive α, and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.

AB - We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form α/x, α > 0. We establish the exponential stability of the semigroup for all positive α, and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.

KW - damped wave equation

KW - singular damping

KW - empty spectrum

KW - finite-time extinction

KW - Laguerre polynomials

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The damped wave equation with singular damping

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Keywords

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