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

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.