The asymptotic behavior of the linear transmission problem in viscoelasticity

Margareth Alves; Jaime Muñoz Rivera; Mauricio Sepúlveda; Octavio Vera Villagrán; María Zegarra Garay

doi:10.1002/mana.201200319

The asymptotic behavior of the linear transmission problem in viscoelasticity

Margareth Alves, Jaime Muñoz Rivera, Mauricio Sepúlveda, Octavio Vera Villagrán, María Zegarra Garay

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

43 Scopus citations

Abstract

We consider a transmission problem with localized Kelvin-Voigt viscoelastic damping. Our main result is to show that the corresponding semigroup (SA(t))t≥0 is not exponentially stable, but the solution of the system decays polynomially to zero as 1/t2 when the initial data are taken over the domain D(A). Moreover, we prove that this rate of decay is optimal. Finally, using a second order scheme that ensures the decay of energy (Newmark-β method), we give some numerical examples which demonstrate this polynomial asymptotic behavior.

Original language	English
Pages (from-to)	483-497
Number of pages	15
Journal	Mathematische Nachrichten
Volume	287
Issue number	5-6
DOIs	https://doi.org/10.1002/mana.201200319
State	Published - Apr 2014
Externally published	Yes

Keywords

Kelvin-Voigt
Newmark-β method
Polynomial decay
Semigroup
Transmission problem
Viscoelastic damping

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10.1002/mana.201200319

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title = "The asymptotic behavior of the linear transmission problem in viscoelasticity",

abstract = "We consider a transmission problem with localized Kelvin-Voigt viscoelastic damping. Our main result is to show that the corresponding semigroup (SA(t))t≥0 is not exponentially stable, but the solution of the system decays polynomially to zero as 1/t2 when the initial data are taken over the domain D(A). Moreover, we prove that this rate of decay is optimal. Finally, using a second order scheme that ensures the decay of energy (Newmark-β method), we give some numerical examples which demonstrate this polynomial asymptotic behavior.",

keywords = "Kelvin-Voigt, Newmark-β method, Polynomial decay, Semigroup, Transmission problem, Viscoelastic damping",

author = "Margareth Alves and Rivera, {Jaime Mu{\~n}oz} and Mauricio Sep{\'u}lveda and Villagr{\'a}n, {Octavio Vera} and Garay, {Mar{\'i}a Zegarra}",

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T1 - The asymptotic behavior of the linear transmission problem in viscoelasticity

AU - Alves, Margareth

AU - Rivera, Jaime Muñoz

AU - Sepúlveda, Mauricio

AU - Villagrán, Octavio Vera

AU - Garay, María Zegarra

PY - 2014/4

Y1 - 2014/4

N2 - We consider a transmission problem with localized Kelvin-Voigt viscoelastic damping. Our main result is to show that the corresponding semigroup (SA(t))t≥0 is not exponentially stable, but the solution of the system decays polynomially to zero as 1/t2 when the initial data are taken over the domain D(A). Moreover, we prove that this rate of decay is optimal. Finally, using a second order scheme that ensures the decay of energy (Newmark-β method), we give some numerical examples which demonstrate this polynomial asymptotic behavior.

AB - We consider a transmission problem with localized Kelvin-Voigt viscoelastic damping. Our main result is to show that the corresponding semigroup (SA(t))t≥0 is not exponentially stable, but the solution of the system decays polynomially to zero as 1/t2 when the initial data are taken over the domain D(A). Moreover, we prove that this rate of decay is optimal. Finally, using a second order scheme that ensures the decay of energy (Newmark-β method), we give some numerical examples which demonstrate this polynomial asymptotic behavior.

KW - Kelvin-Voigt

KW - Newmark-β method

KW - Polynomial decay

KW - Semigroup

KW - Transmission problem

KW - Viscoelastic damping

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DO - 10.1002/mana.201200319

M3 - Artículo

AN - SCOPUS:84897422775

SN - 0025-584X

VL - 287

SP - 483

EP - 497

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

IS - 5-6

ER -

The asymptotic behavior of the linear transmission problem in viscoelasticity

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Keywords

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