In this study, we consider a viscoelastic wave equation in the presence of finite and infinite memories. Under suitable conditions on the variable coefficients and for a wide class of relaxation functions, we establish an explicit and general decay result. Moreover, we use a weaker boundedness condition, on the initial data than the one used in the literature. The proof of our decay result is based on the multiplier method together with some convexity arguments. This study generalizes and improves previous literature outcomes.

1. Introduction

Let us consider an -dimensional body occupies with a bounded open set with smooth boundary . Let be the position of the material particle at time so that ; the corresponding motion equation is

The functions and are called the relaxation (kernels) functions, and they are positive nonincreasing and defined on . The functions and are essentially bounded nonnegative defined on . Here, and are the history and initial data. This model of materials consist of an elastic part (without memory) and a viscoelastic part, where the dissipation given by the memory is effective.

In this study, we are concerned with the above viscoelastic wave problem (1) and mainly interested in the asymptotic behavior of the solution when tends to infinity. In fact, we prove that the solution of the corresponding viscoelastic model decays 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 to prescribe the history of before 0. Here, we assume that . Let us mention some other papers related to the problem 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 goes to infinity. However, no rate of decay has been specified. In [2], a similar result, under a convexity 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. For example, Hrusa [3] considered a one-dimensional nonlinear viscoelastic equation of the formand proved several global existence results for large data. He also proved an exponential decay for strong solutions when and satisfies certain conditions. Dassios and Zafiropoulos [4] studied a viscoelastic problem in 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 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 [6], 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.

Decay results for arbitrary growth of the damping term have been considered for the first time in the work of Lasiecka and Tataru [8]. They showed that the energy decays as fast as the solution of an associated differential equation whose coefficients depend on the damping term. Following the method of Lasiecka and Tataru [8], Liu and Zuazua [9] established decay rates for a nonlinear globally distributed damping for the wave equation with no growth assumptions at the origin. The proof of their result is based on constructing a convex function which captures the nonlinearity of the feedback at the origin and combines it, using Jensen’s and Young’s inequalities, with the use of equivalent energy which satisfies a differential equation, which however is not a priori dissipative. Alabau-Boussouira [10] used weighted integral inequalities and some convexity arguments to establish a semiexplicit formula which leads to decay rates of the energy in terms of the behavior of the nonlinear feedback close to the origin, for which the optimal exponential and polynomial decay rate estimates are only special cases. Alabau-Boussouira [11] presented a general approach based on convexity arguments to establish sharp optimal or quasi-optimal upper energy decay rates for finite and infinite dimensional vibrating damped systems.

Inspired by the experience with frictional damping initiated in the work of Lasiecka and Tataru [8], another step forward was done by Alabau-Boussouira and Cannaras [12] when they considered relaxation functions satisfying

They established an explicit rate of decay under some constraints imposed on the function .

For partially viscoelastic material, Rivera and Salvatierra [13] showed that the energy decays exponentially and provided 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 [14]. The uniform decay of solutions for the viscoelastic wave equation,was investigated by Cavalcanti and Oquendo [15] where they considered the condition . They established exponential and polynomial stability results based on some conditions on and the linearity of the function . After that, Guesmia and Messaoudi [16] extended the work of [15], and they established a general decay result for (1) under the same conditions on and used in [15] and for some other conditions for the relaxation functions and , from which the usual exponential and polynomial decay rates are only special cases. More precisely, they used the conditions , andsuch thatwhere is an increasing strictly convex function.

In the present work, we consider (1) and under the same conditions on and used in [15], and for a large class of the relaxation functions, we prove that problem (1) is stable. Under this class of relaxation functions at infinity, we establish a relation between the decay rate of the solution and the growth of at infinity. In fact, we extend the works of [15, 16]. More precisely, we assume that the relaxations functions and satisfy

In addition, we use a weaker boundedness condition on the history data used in earlier papers such as [1720].

The paper is organized as follows. In Section 2, we present some material needed for our work. We establish some essential lemmas in Section 3. Section 4 contains the statement and the proof of our main result. We give some examples in Sections 5 to illustrate our energy decay result. Finally, we introduce conclusions and comparison study in Section 6.

2. Preliminaries

In this section, we present some material needed in the proof of our main result. Through this study, we use to denote a positive generic constant. Now, we start with the following assumptions:(A1) are differentiable nonincreasing functions such that, for any constant and ,and there exist functions which are linear or strictly increasing and strictly convex functions on for some with , , , and are convex on . Moreover, there exist positive nonincreasing differentiable functions , such that(A2) are functions such that, for positive constants and and for with meas and(A3) We assume that

Remark 1. The class of relaxation functions satisfying in the present study is larger than the ones satisfying (5) and (6) used in [16]. In fact, the boundedness of the sup in (5), used in [16], can be interpreted as the inequality in in the present paper (with ). Conditions (5) and (6) used in [16] ask also the boundedness of the integral. Moreover, the classical of relaxation functions that we consider in our example , satisfy both conditions in our paper and the one in [16]. So, it is better to consider used in the present study than the one used in [16].

Remark 2. Hypothesis is needed for proving the existence and stability results. For the stability, if holds, then which is defined in (25) is well defined, so inequality (53) makes sense. Moreover, Hypothesis is weaker than the one used in [16, 18]; that is, there exists a positive constant such thatand our decay rate depends on the size of .

Remark 3. If , there exist neighborhoods of , such thatAs in [15, 16], let and let , be such that

Lemma 1. The functions , are not identically zero and satisfy

Proof. (1)For , we have , which implies, by (14), that . Thus, is not identically zero.(2)If , then . Consequently, . If , then which implies, by (14), . Consequently, . This completes the proof.The existence and uniqueness of the solution of problem (1) is given by the following theorem.

Theorem 1. If and hypotheses hold, then problem (1) has a unique solution in the class

The proof of the above theorem can be obtained, making use, for instance, of the Faedo–Galerkin method and considering standard arguments of density (see [15]).

We define the “modified” energy functional of the weak solution of problem (1) bywhere

Lemma 2. The “modified” energy functional satisfies, along the solution of problem (1), the following:where

Proof. By multiplying the first equation in problem (1) by and integrating over , using integration by parts, hypotheses and and some manipulations as in [6,21] and others, we obtain (19) for regular solutions. This inequality remains valid for weak solutions by a simple density argument.

3. Technical Lemmas

In this section, we present and prove additional lemmas that are pivotal for the principal result.

Lemma 3 (see [22]). For , we havewhere, for any ,

Furthermore, using the fact thatand recalling the Lebesgue dominated convergence theorem, one can deduce that

Lemma 4. Assume that hold. There exists a positive constant such thatwhere .

Proof. The proof is identical to the one in [23]. Indeed, we havewhere .

Lemma 5 (see [16]). Assume that hold; then, the functionals,satisfy, along the solution of (1) and for any and ; the following estimatesand

Lemma 6. Assume that hold; then, for a suitable choice of and for all , the functional,satisfies, for any

Proof. Let, for fixed, . Then, direct differentiation of and using (19), (28), (29), and (30) leads toRecalling that , where . We obtainSimplifying the above estimate, we arrive atBy using the fact that and choosingwe obtainAs a consequence of (24), there exists such that if , thenIt is clear that (38) givesChoosing , then, for some , we haveThen, (32) is established. Moreover, one can choose large enough so that .

Lemma 7. Assume that hold. Then, for all and fixed positive constant , we have the following estimates:where is defined in (25).

Proof. The proof of this lemma can be established by following the same arguments in [23, 24].

Lemma 8. If are satisfied, then we have, for all and for , the following estimates:where , , and are introduced in , and

Proof. To establish (42), we introduce the following functional:Then, using (17) and the fact that is nonincreasing to get,Using (41) and the definition of , (45) becomesTherefore, we can choose such that, for all ,Without loss of the generality, for all , we assume that ; otherwise, we get an exponential decay from (32). The use of Jensen’s inequality and using (43) and (47) giveHence, (42) is established.

4. Decay Result

In this section, we state and prove our main result and provide some examples to illustrate our decay results. Let us start introducing some functions and then establishing several lemmas needed for the proof of our main result. Now, for , let us take

As in [23], we introduce the following functions:where is the convex conjugate of in the sense of Young (see [25]). One can easily verify that is decreasing function over , and , and are convex and increasing functions on . Furthermore, we introduce the class of functions satisfying for fixed :andwhere is defined in (25) and .

Theorem 2. Assume that hold; then, there exists a strictly positive constant such that the solution of (1) satisfies, for all ,where and are defined in (50) and (52), respectively.

Proof. Using (25), (32), and (42), then, for some positive constant and any , we obtainWithout loss of generality, one can assume that . For , let the functional be defined bywhich satisfies . By noting that , , and , we obtainLet be the convex conjugate of in the sense of Young (see [25]); then,and satisfies the following generalized Young inequality:So, with and and using (57)–(59), we arrive atSo, multiplying (60) by and using (43) and the fact that giveConsequently, recalling the definition of and choosing so that , we obtain, for all ,where and satisfies for some :Since and is strictly increasing and strictly convex on , we find that on . Using the general Young inequality (59) on the last term in (62) with and , we have, for ,Now, combining (62) and (64) and choosing small enough so that , we arrive atUsing the equivalent property in (63) and the increasing of , we haveLetting and recalling , then we arrive at, for some ,Since is nonincreasing, using the equivalent property implies that there exists such that . Let and satisfying (52) and (53). Ifthen, we haveIfthen, for any , we obtainsince is a nonincreasing function. Therefore, for any , we haveUsing (72), recalling the definition of , the fact that is convex, and , we have, for any and ,Now, we letwhere small enough so that . Then, (73) becomes, for any ,Furthermore, we haveSince and using (67), then, for any , , we obtainThen, using (53) and (75), we obtainFrom the definition of and , we haveHence,Now, we haveThen, according to (53), we obtainThen, (78) givesThus, from (83) and the definition of and in (50) and (51), we obtainIntegrating (84) over , we obtainSince is decreasing, and , thenRecalling that , we haveSimilarly, recall that ; then,Since , then, for some