Let us consider an $n$-dimensional body occupies with a bounded open set $\Omega\subseteq\mathbb{R}^{n}(n\geq 1)$ with smooth boundary $\partial\Omega$. Let $x\rightarrow u(x,t)$ be the position of the material particle $x$ at time $t$. So that, $\forall x\in\Omega,\forall t>0$, the corresponding motion equation is

$$\begin{array}[]{ll}u_{tt}-\nabla u+\int_{0}^{t}g_{1}(t-s)div(a_{1}(x)\nabla u(s))ds+\int_{0}^{+\infty}g_{2}(s)div(a_{2}(x)\nabla u(t-s))ds=0,\\ u(x,t)=0,\\ u(x,-t)=u_{0}(x,t),u_{t}(x,0)=u_{1}(x).\end{array}$$

(1.1)

The functions $g_{1}$ and $g_{2}$ are called the relaxation (kernels) functions and they are positive non-increasing and defined on $\mathbb{R}^{+}$. The functions $a_{1}$ and $a_{2}$ are essentially bounded non-negative defined on $\Omega$. Here, $u_{0}$ and $u_{1}$ are the given initial data. This model of materials consisting of an elastic part (without memory) and a viscoelastic, part, where the dissipation given by the memory is effective.
In this paper, we are concerned with the above viscoelastic wave problem (1.1) and mainly interested in the asymptotic behavior of the solution $u$ when $t$ tends to infinity. In fact, We prove that the solutions of the corresponding viscoelastic model decay to zero and no matter how small is the viscoelastic part of the material. Note that the above model is dissipative, and the dissipation is given by the memory term only and the memory is effective only in a part of the body. For materials with memory the stress depends not only on the present values but also on the entire temporal history of the motion. Therefore, we have also to prescribe the history of $u$ before 0. Here, we assume that $u(x,-t)=u_{0}(x,t),\forall x\in\Omega,\forall t>0.$ Let us mention some other papers related to the problems we address. We start our literature review with the pioneer work of Dafermos [1], in 1970, where the author discussed a certain one-dimensional viscoelastic problem, established some existence results, and then proved that, for smooth monotone decreasing relaxation functions, the solutions go to zero as $t$ goes to infinity. However, no rate of decay has been specified. In Dafermos [2], a similar result, under a ity condition on the kernel, has been established. After that a great deal of attention has been devoted to the study of viscoelastic problems and many existence and long-time behavior results have been established. Hrusa [3] considered a one-dimensional nonlinear viscoelastic equation of the form

$$u_{tt}-cu_{xx}+\int_{0}^{t}m(t-s)\left(\psi(u_{x}(x,s))\right)_{x}~{}ds=f(x,t)$$

(1.2)

and proved several global existence results for large data. He also proved an exponential decay for strong solutions when $m(s)=e^{-s}$ and $\psi$ satisfies certain conditions. Dassios and Zafiropoulos [4] studied a viscoelastic problem in $\mathbb{R}^{3}$ and proved a polynomial decay results for exponentially decaying kernels. After that, a very important contribution by Rivera was introduced. In 1994, Rivera [5] considered equations for linear isotropic homogeneous viscoelastic solids of integral type which occupy a bounded domain or the whole space $\mathbb{R}^{n}$. in the bounded domain case, and for exponentially decaying memory kernel and regular solutions, he showed that the sum of the first and the second energy decays exponentially. For the whole-space case and for exponentially decaying memory kernels, he showed that the rate of decay of energy is of algebraic type and depends on the regularity of the solution. this result was later generalized to a situation, where the kernel is decaying algebraically but not exponentially by Cabanillas and Rivera [6]. In the paper, the authors considered the case of bounded domains as well as the case when the material is occupying the entire space and showed that the decay of solutions is also algebraic, at a rate which can be determined by the rate of decay of the relaxation function. This latter result was later improved by Baretto et al. [7], where equations related to linear viscoelastic plates were treated. Precisely, they showed that the solution energy decays at the same rate of the relaxation function. For partially viscoelastic material, Rivera and Salvatierra [8] showed that the energy decays exponentially, provide the relaxation function decays in a similar fashion and the dissipation is acting on a part of the domain near to the boundary. See also, in this direction, the work of Rivera and Oquendo [9]. The uniform decay of solutions for the viscoelastic wave equation

$$u_{tt}-k_{0}\nabla u+\int_{0}^{t}div\left[a(x)g(t-\tau)\nabla u(\tau)\right]d\tau+b(x)h(u_{t})+f(u)=0,$$

(1.3)

was investigated by Cavalcanti and Oquendo [10] where they considered the condition $a(x)+b(x)\geq\delta>0$. They established exponential and polynomial stability results based on some conditions on $g$ and the linearity of the function $h$. After that, Guesmiaa and Messaoudib [11] extended the work of [10] and they establish a general decay result for (1.1) under the same conditions on $a_{1}$ and $a_{2}$ used in [10] and for some other conditions for the relaxation functions $g_{1}$ and $g_{2}$, from which the usual exponential and polynomial decay rates are only special cases. More precisely, they used the conditions $g_{1}^{\prime}(t)\leq-\xi(t)g_{1}(t),\hskip 10.84006pt\forall t\geq 0,$ and

$$\int_{0}^{+\infty}\frac{g_{2}(s)}{G^{-1}(-g_{2}^{\prime}(s))}ds+\sup_{s\in\mathbb{R}_{+}}\frac{g_{2}(s)}{G^{-1}(-g_{2}^{\prime}(s))}<+\infty,$$

(1.4)

such that

$$G(0)=G^{\prime}(0)=0\hskip 3.61371pt\text{and}\hskip 3.61371pt\lim_{t\rightarrow+\infty}{G^{\prime}(t)}=+\infty,$$

(1.5)

where $G:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is an increasing strictly convex function.
In the present work, we extend the works of [10] and [11]. In fact, we consider (1.1) and under the same conditions on $a_{1}$ and $a_{2}$ used in [10] and for a large class of the relaxation functions, we prove a general decay result. More precisely, we assume that the relaxations functions $g_{1}$ and $g_{2}$ are satisfying

$$g_{i}^{\prime}(t)\leq-\xi(t)H_{i}(g_{i}(t)),\hskip 10.84006pt\forall t\geq 0.$$

In fact, our result allows a large class of relaxation functions and improves the decay rates in some earlier papers. The paper is organized as follows. In Section 2, we present some material needed for our work. Some essential lemmas are presented and established in Section 3. Section 4 contains the statement and the proof of our main result. We end our paper by giving some illustrating examples in Sections 5.

In this section, we present some material needed in the proof of our main result. Through this paper, we use $c$ to denote a positive generic constant. Now, we start with the following assumptions:
(A1) $g_{i}:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ are differentiable non-increasing functions such that

$$g_{i}(0)>0,~{}~{}i=1,2,~{}~{}~{}~{}~{}1-||a_{1}||_{\infty}\int_{0}^{+\infty}g_{1}(s)ds-||a_{2}||_{\infty}\int_{0}^{+\infty}g_{2}(s)ds=l>0.$$

and there exists $C^{1}$ functions $H_{i}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ which are linear or it is strictly increasing and strictly convex $C^{2}$ function on $(0,r]$ for some $r>0$ with $H_{i}(0)=H_{i}^{\prime}(0)=0$, such that

$$g_{i}^{\prime}(t)\leq-\xi(t)H_{i}(g_{i}(t)),\hskip 10.84006pt\forall t\geq 0,$$

where $\xi_{i}:\mathbb{R}^{+}\longrightarrow\mathbb{R}^{+}$ are positive nonincreasing differentiable functions.
(A2) $a_{i}:\bar{\Omega}\longrightarrow\mathbb{R}^{+}$ are in $C^{1}(\bar{\Omega})$ such that, for positive constant $\delta$ and $a_{0}$ and for $\Gamma_{1},\Gamma_{2}\subset\partial\Omega$ with means $(\Gamma_{i})>0,~{}i=1,2,~{}\inf_{x\in\bar{\Omega}}(a_{1}(x)+a_{2}(x))\geq\delta$ and

$$a_{i}=0~{}~{}~{}\text{or}~{}~{}~{}\inf_{\Gamma_{i}}~{}a_{i}(x)\geq 2a_{0},~{}~{}~{}~{}~{}~{}i=1,2.$$

As in [10, 11], let $d=\min\{a_{0},\delta\}$ and let $\alpha_{i}\in C^{1}(\bar{\Omega}),~{}i=1,2,$ be such that

$$\begin{cases}\begin{array}[]{ll}0\leq\alpha_{i}(x)\leq a_{i}(x)&\\ \alpha_{i}(x)=0,&\text{if}~{}~{}a_{i}(x)\leq\frac{d}{4}\\ \alpha_{i}(x)=a_{i}(x),&\text{if}~{}~{}a_{i}(x)\geq\frac{d}{2}.\\ \end{array}\end{cases}$$

(2.1)

The existence and uniqueness of the solution of problem (1.1) can be established by using the Galerkin method. We define the ”modified” energy functional of the weak solution of problem (1.1), by

$$\begin{array}[]{c}E(t)=\frac{1}{2}\int_{\Omega}u_{t}^{2}\partial x+\frac{1}{2}\int_{\Omega}\left[1-a_{1}(x)\int_{0}^{t}g_{1}(s)ds-a_{2}(x)\int_{0}^{+\infty}g_{2}(s)ds\right]|\nabla u|^{2}dx\\ +\frac{1}{2}~{}g_{1}~{}o~{}\nabla u+\frac{1}{2}~{}g_{2}~{}o~{}\nabla u,\end{array}$$

(2.2)

where

$$\begin{array}[]{l}g_{1}~{}o~{}\nabla u=\int_{\Omega}a_{1}(x)\int_{0}^{t}g_{1}(t-s)|\nabla u(t)-\nabla u(s)|^{2}ds~{}dx\\ g_{2}~{}o~{}\nabla u=\int_{\Omega}a_{2}(x)\int_{0}^{+\infty}g_{2}(s)|\nabla u(t)-\nabla u(t-s)|^{2}ds~{}dx\\ .\end{array}$$

(2.3)

In this section, we introduce some fundamental lemmas. These lemmas will help us for proving our results. We use our equations and assumptions to prove those lemmas.

Furthermore, using the fact that

$$\frac{\alpha g_{i}^{2}(s)}{\alpha g_{i}(s)-g_{i}^{\prime}(s)}<g_{i}(s)$$

and recalling the Lebesgue dominated convergence theorem, one can deduce that

$$\alpha C_{\alpha}=\int_{0}^{\infty}\frac{\alpha g_{i}^{2}(s)}{\alpha g_{i}(s)-g_{i}^{\prime}(s)}ds\to 0\text{ as }\alpha\to 0.$$

(3.3)