Stability result for a viscoelastic wave equation in the presence of finite and infinite memories

TL;DR

This paper proves a general decay stability result for a viscoelastic wave equation with finite and infinite memory effects, removing previous restrictions on initial data and extending existing theories.

Contribution

It introduces a new decay result for the viscoelastic wave equation with both finite and infinite memories, generalizing and improving prior results.

Findings

Established a general decay stability result.

Removed restrictions on initial data boundedness.

Extended previous literature on viscoelastic wave equations.

Abstract

In this paper, we are concerned with the following viscoelastic wave equation \begin{equation*} \label{1} u_{tt}-\nabla u +\int_0^t g_1 (t-s)~ div(a_1(x) \nabla u(s))~ ds + \int_0^{+ \infty} g_2 (s)~ div(a_2(x) \nabla u(t-s)) ~ds = 0, \end{equation*} in a bounded domain $\Omega$. Under suitable conditions on $a_1$ and $a_2$ and for a wide class of relaxation functions $g_1$ and $g_2$. We establish a general decay result. The proof is based on the multiplier method and makes use of convex functions and some inequalities. More specifically, we remove the constraint imposed on the boundedness condition on the initial data $\nabla u_{0}$. This study generalizes and improves previous literature outcomes.