# American Institute of Mathematical Sciences

January  2021, 20(1): 389-404. doi: 10.3934/cpaa.2020273

## 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

Received  November 2019 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: This work is funded by KFUPM under Project #SB191037

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.
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
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