Abstract

In this paper, we prove a general energy decay results of a coupled Lamé system with distributed time delay. By assuming a more general of relaxation functions and using some properties of convex functions, we establish the general energy decay results to the system by using an appropriate Lyapunov functional.

1. Introduction

In this work, we shall be concerned with studying the general decay rate of the following Lamé system in :

Equations (1) are associated with the following boundary and initial conditions where is a bounded domain in , with smooth boundary . The elasticity differential operator is given by and the Lamé constants and are satisfying the following conditions

The parameters , and are positive constants, with . The functions are bounded. The functions and which represent the source terms will be specified later.

After several authors have studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities and is proven under conditions imposed on the subgroup [1]. The researchers also studied behavior of the energy in a limited field with nonlinear damping and external force and a varying delay of time to find solutions to the Lame system [19].

Recently, problems that contain viscoelasticity have been addressed, and many results have been found regarding the global existence and stability of solutions (see [2, 9]), under conditions on the relaxation function, whether exponential or polynomial decay. In addition, in [10], Boulaaras obtained the stability result of the global solution to the Lamé system with the flexible viscous term by adding logarithmic nonlinearity, even though the kernel is not necessarily decreasing in contrast to what he studied [2].

Introducing a distributed delay term makes our problem different from those considered so far in the literature.

The importance of this term appears in many works, and this is due to the fact that many phenomena depends on their past. Also, it is influence on the asymptotic behavior of the solution for the different types of problems such that Timoshenko system [3, 1113], transmission problem [14], wave equation [15], and thermoelastic system [16, 17].

In the present work, we extend the general decay result obtained by Feng in [18] to the case of distributed term delay, namely, we will make sure that the result is achieved if the distributed delay term exists.

This paper is organized as follows. In the second section, we give some preliminaries related to problem (1). In Section 3, we prove our main result.

2. Preliminaries

In this section, we provide some materials and necessary assumptions which we need in the prove of our results. We use the standard Lebesgue and Sobolev spaces with their scaler products and norms. For simplicity, we would write instead of . Throughout this work, we used a generic positive constant .

For the relaxation functions and , we assume, for ,

(A1) are nonincreasing functions satisfying

We assume further that for

(A2) There exist two functions , with , which are linear or are strictly increasing and strictly convex functions of class on , such that where are functions satisfying

(A3) For the source terms and , we take with . Clearly, where Further, we assume that there is , such that

(A4)

So, we have the embedding

Let the same embedding constant, so we have

Remark 1. There exist two constants and such that

As in many papers, we introduce the following new variables then, we obtain

Consequently, the problem (1) is equivalent to with the initial data and boundary conditions where

We recall the following notations

Thus, we have the following important property

The energy modified associated to the problem (19) is defined by

First, we prove in the following theorem the result of energy identity.

Lemma 2. Assume that

Then, the energy modified defined by (24) satisfies, along the solution of (19), the estimate for

Proof. First multiplying the equation by and integrating by parts over we obtain by using (23), we obtain Similarly, multiplying the equation (19) by and integrating over we obtain Multiplying the equation (19) by and integrating by parts over we obtain therefore Multiplying the fourth equation of (19) by and integrating over we obtain For the source term, we have By collecting the previous equations (29)–(34), we get Using Young’s inequality, we obtain similarly This completes the proof.

3. General Decay

In this section we will prove that the solution of problems (19)–(20) decay generally to trivial solution. Using the energy method and suitable Lyapunov functional.

In the following, we will present our main stability result:

Theorem 3 (Decay rates of energy). Assume that (A1)(A3) hold. Then, for every , there exist two positive constants and such that the energy defined by (24) satisfies the following decay where and

This theorem will be proved later after providing some remarks.

Remark 4. (1)In case , Theorem 3 ensures .(2)From (A2), we infer that . Then, there exists some large enough such that (a)As is positive continuous functions, and and are positive nonincreasing continuous functions, then, for all which implies for some positive constants and , Consequently, (3)We also mention Johnson’s inequality, which is very important for proving our result. If is a convex function on , , we havewhere is a function that satisfies

To prove the desired result, we create a Lyapunov functional equivalent to . For this, we define some functions that allow us to construct this Lyapunov function.

As in Baowei [18] and Mustafa ([19, 20]), we define for any .

Lemma 5. Let be a solution of the problem (19). Then, the functional satisfies the estimate

Proof. Taking the derivative of (47), we obtain From problem (19) and using integration by parts, we get By using Hölder and Young’s inequalities, we have Similarly, we obtain The Young’s inequality gives For the source term, we have Combining the equations (51)–(54), thus, our proof is completed.

Lemma 6. Let (, , , ) be a solution of the problem (19). Then, the functional satisfies for any the estimate where and are two positive constants.

Proof. First, we begin to estimate