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

Article Contents

2021, Volume 20, Issue 1: 389-404. Doi: 10.3934/cpaa.2020273

This issue Previous Article On optimal autocorrelation inequalities on the real line Next Article Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials

New general decay result for a system of viscoelastic wave equations with past history

1.
The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2.
Department of Mathematics, University of Sharjah, P. O. Box, 27272, Sharjah. UAE

^*Corresponding author
^*Corresponding author

Received: November 2019

Revised: September 2020

Early access: November 15, 2020

Published online: November 15, 2020

This work is funded by KFUPM under Project #SB191037.

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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 $ g_i : [0, +\infty) \rightarrow (0, +\infty) $ satisfying

$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $

where $ \xi_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.

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Mathematics Subject Classification: Primary: 35B35, 35B40, 93D20.

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