NEW GENERAL DECAY RESULT for A SYSTEM of VISCOELASTIC WAVE EQUATIONS with PAST HISTORY

Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Salim A. Messaoudi

doi:10.3934/cpaa.2020273

NEW GENERAL DECAY RESULT for A SYSTEM of VISCOELASTIC WAVE EQUATIONS with PAST HISTORY

Adel M. Al-Mahdi^*
, Mohammad M. Al-Gharabli
, Salim A. Messaoudi

^*Corresponding author for this work

Preparatory Year Program

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

9 Scopus citations

Abstract

This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels gi : [0, +∞) → (0, +∞) satisfying g_i⁰(t) ≤ −ξi(t)H_i(gi(t)), ∀ t ≥ 0 and for i = 1, 2, where ξ_i and H_i are functions satisfying some specific properties. Under this very general assumption on the behavior of g_i at infinity, we establish a relation between the decay rate of the solutions and the growth of g_i at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

Original language	English
Pages (from-to)	389-404
Number of pages	16
Journal	Communications on Pure and Applied Analysis
Volume	20
Issue number	1
DOIs	https://doi.org/10.3934/cpaa.2020273
State	Published - Jan 2021

Bibliographical note

Publisher Copyright:
© 2021 American Institute of Mathematical Sciences. All rights reserved.

Keywords

And phrases. Infinite memory
Convex functions
General decay
Relaxation function
Viscoelastic equations

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.3934/cpaa.2020273

Cite this

@article{36f7214de70e438187d9d59456b79180,

title = "NEW GENERAL DECAY RESULT for A SYSTEM of VISCOELASTIC WAVE EQUATIONS with PAST HISTORY",

abstract = "This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels gi : [0, +∞) → (0, +∞) satisfying gi0(t) ≤ −ξi(t)Hi(gi(t)), ∀ t ≥ 0 and for i = 1, 2, where ξi and Hi are functions satisfying some specific properties. Under this very general assumption on the behavior of gi at infinity, we establish a relation between the decay rate of the solutions and the growth of gi at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.",

keywords = "And phrases. Infinite memory, Convex functions, General decay, Relaxation function, Viscoelastic equations",

author = "Al-Mahdi, \{Adel M.\} and Al-Gharabli, \{Mohammad M.\} and Messaoudi, \{Salim A.\}",

year = "2021",

month = jan,

doi = "10.3934/cpaa.2020273",

language = "English",

volume = "20",

pages = "389--404",

journal = "Communications on Pure and Applied Analysis",

issn = "1534-0392",

publisher = "American Institute of Mathematical Sciences",

number = "1",

}

TY - JOUR

T1 - NEW GENERAL DECAY RESULT for A SYSTEM of VISCOELASTIC WAVE EQUATIONS with PAST HISTORY

AU - Al-Mahdi, Adel M.

AU - Al-Gharabli, Mohammad M.

AU - Messaoudi, Salim A.

PY - 2021/1

Y1 - 2021/1

N2 - This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels gi : [0, +∞) → (0, +∞) satisfying gi0(t) ≤ −ξi(t)Hi(gi(t)), ∀ t ≥ 0 and for i = 1, 2, where ξi and Hi are functions satisfying some specific properties. Under this very general assumption on the behavior of gi at infinity, we establish a relation between the decay rate of the solutions and the growth of gi at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

AB - This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels gi : [0, +∞) → (0, +∞) satisfying gi0(t) ≤ −ξi(t)Hi(gi(t)), ∀ t ≥ 0 and for i = 1, 2, where ξi and Hi are functions satisfying some specific properties. Under this very general assumption on the behavior of gi at infinity, we establish a relation between the decay rate of the solutions and the growth of gi at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

KW - And phrases. Infinite memory

KW - Convex functions

KW - General decay

KW - Relaxation function

KW - Viscoelastic equations

UR - https://www.scopus.com/pages/publications/85097013096

U2 - 10.3934/cpaa.2020273

DO - 10.3934/cpaa.2020273

M3 - Article

AN - SCOPUS:85097013096

SN - 1534-0392

VL - 20

SP - 389

EP - 404

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

IS - 1

ER -

NEW GENERAL DECAY RESULT for A SYSTEM of VISCOELASTIC WAVE EQUATIONS with PAST HISTORY

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Fingerprint

Cite this