Exponential and polynomial decay for a quasilinear viscoelastic equation

Salim A. Messaoudi; Nasser eddine Tatar

doi:10.1016/j.na.2006.11.036

Exponential and polynomial decay for a quasilinear viscoelastic equation

Salim A. Messaoudi^*
, Nasser eddine Tatar

^*Corresponding author for this work

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

123 Scopus citations

Abstract

We consider the following nonlinear viscoelastic equation: | u_t |^ρ u_{t t} - Δ u - Δ u_{t t} + ∫₀^t g (t - s) Δ u (s) d s = 0, in a bounded domain Ω for ρ > 0. We show that the dissipation induced by the integral term is strong enough to stabilize the solution. This result improves an earlier one given by Cavalcanti et al. in [M.M. Cavalcanti, V.N.D. Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043-1053].

Original language	English
Pages (from-to)	785-793
Number of pages	9
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	68
Issue number	4
DOIs	https://doi.org/10.1016/j.na.2006.11.036
State	Published - 15 Feb 2008

Bibliographical note

Funding Information:
The authors would like to express their sincere thanks to King Fahd University of Petroleum and Minerals for its support. This work was funded by KFUPM under Project # SABIC 2005-02.

Keywords

Exponential decay
Memory term
Modified energy functional
Nonlinear viscoelastic equation
Polynomial decay
Relaxation function

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2006.11.036

Cite this

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title = "Exponential and polynomial decay for a quasilinear viscoelastic equation",

abstract = "We consider the following nonlinear viscoelastic equation: | ut |ρ ut t - Δ u - Δ ut t + ∫0t g (t - s) Δ u (s) d s = 0, in a bounded domain Ω for ρ > 0. We show that the dissipation induced by the integral term is strong enough to stabilize the solution. This result improves an earlier one given by Cavalcanti et al. in [M.M. Cavalcanti, V.N.D. Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043-1053].",

keywords = "Exponential decay, Memory term, Modified energy functional, Nonlinear viscoelastic equation, Polynomial decay, Relaxation function",

author = "Messaoudi, \{Salim A.\} and Tatar, \{Nasser eddine\}",

note = "Funding Information: The authors would like to express their sincere thanks to King Fahd University of Petroleum and Minerals for its support. This work was funded by KFUPM under Project \# SABIC 2005-02.",

year = "2008",

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day = "15",

doi = "10.1016/j.na.2006.11.036",

language = "English",

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journal = "Nonlinear Analysis, Theory, Methods and Applications",

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TY - JOUR

T1 - Exponential and polynomial decay for a quasilinear viscoelastic equation

AU - Messaoudi, Salim A.

AU - Tatar, Nasser eddine

N1 - Funding Information: The authors would like to express their sincere thanks to King Fahd University of Petroleum and Minerals for its support. This work was funded by KFUPM under Project # SABIC 2005-02.

PY - 2008/2/15

Y1 - 2008/2/15

N2 - We consider the following nonlinear viscoelastic equation: | ut |ρ ut t - Δ u - Δ ut t + ∫0t g (t - s) Δ u (s) d s = 0, in a bounded domain Ω for ρ > 0. We show that the dissipation induced by the integral term is strong enough to stabilize the solution. This result improves an earlier one given by Cavalcanti et al. in [M.M. Cavalcanti, V.N.D. Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043-1053].

AB - We consider the following nonlinear viscoelastic equation: | ut |ρ ut t - Δ u - Δ ut t + ∫0t g (t - s) Δ u (s) d s = 0, in a bounded domain Ω for ρ > 0. We show that the dissipation induced by the integral term is strong enough to stabilize the solution. This result improves an earlier one given by Cavalcanti et al. in [M.M. Cavalcanti, V.N.D. Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043-1053].

KW - Exponential decay

KW - Memory term

KW - Modified energy functional

KW - Nonlinear viscoelastic equation

KW - Polynomial decay

KW - Relaxation function

UR - https://www.scopus.com/pages/publications/36749047904

U2 - 10.1016/j.na.2006.11.036

DO - 10.1016/j.na.2006.11.036

M3 - Article

AN - SCOPUS:36749047904

SN - 0362-546X

VL - 68

SP - 785

EP - 793

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 4

ER -

Exponential and polynomial decay for a quasilinear viscoelastic equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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