Polynomial Decay Rate for a Coupled Lamé System with Viscoelastic Damping and Distributed Delay Terms

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.


Introduction
In this work, we shall be concerned with studying the general decay rate of the following Lamé system in Ω × ℝ + : Equations (1) are associated with the following boundary and initial conditions where Ω is a bounded domain in ℝ n ðn = 1, 2, 3Þ, with smooth boundary ∂Ω. The elasticity differential operator Δ e is given by and the Lamé constants μ and λ are satisfying the following conditions The parameters k 1 , k 2 , τ 1 , and τ 2 are positive constants, with τ 1 < τ 2 . The functions μ 1 , μ 2 : ½τ 1 , τ 2 → ℝ are bounded. The functions f 1 ðu, vÞ and f 2 ðu, vÞ 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 [1][2][3][4][5][6][7][8][9].
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,[11][12][13], 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.

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 k:k instead of k:k 2 . Throughout this work, we used a generic positive constant c.
For the relaxation functions g 1 and g 2 , we assume, for i = 1, 2, (A1) g i ðtÞ: ℝ + → ℝ + are nonincreasing C 1 functions satisfying We assume further that for i = 1, 2 : (A2) There exist two C 1 functions G i : ℝ + → ℝ + , with G i ð0Þ = G i ′ð0Þ = 0, which are linear or are strictly increasing and strictly convex functions of class C 2 ðℝ + Þ on ð0, r, r ≤ g i ð0Þ, such that where ξ i ðtÞ are C 1 functions satisfying (A3) For the source terms f 1 and f 2 , we take with α, β > 0. Clearly, Further, we assume that there is C > 0, such that So, we have the embedding Let c s the same embedding constant, so we have Remark 1. There exist two constants Λ 1 > 0 and Λ 2 > 0 such that As in many papers, we introduce the following new variables , we obtain Journal of Function Spaces 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 ðu, v, z, yÞ of (19), the estimate for Proof. First multiplying the equation ð0:14Þ 1 by u t and integrating by parts over Ω, we obtain by using (23), we obtain

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 t 0 > 0, there exist two positive constants α 1 and α 2 such that the energy defined by (24) satisfies the following decay where and ξðtÞ = min fξ 1 ðtÞ, ξ 2 ðtÞg, gðtÞ = max fg 1 ðtÞ, g 2 ðtÞg: This theorem will be proved later after providing some remarks.
(2) From (A2), we infer that lim t→∞ g i ðtÞ = 0. Then, there exists some t 1 ≥ 0 large enough such that (a) As G i is positive continuous functions, and g i and ξ i are positive nonincreasing continuous functions, then, for all 0 ≤ t ≤ t 1 , which implies for some positive constants a i and b i , Consequently, (3) We also mention Johnson's inequality, which is very important for proving our result. If G is a convex function on ½a, b, g : Ω → a, b, we have where h is a function that satisfies To prove the desired result, we create a Lyapunov functional equivalent to E. For this, we define some functions that allow us to construct this Lyapunov function.

Lemma 5.
Let ðu, v, z, yÞ be a solution of the problem (19). Then, the functional satisfies the estimate Proof. Taking the derivative of (47), we obtain

Journal of Function Spaces
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 (u, v, z, y) be a solution of the problem (19). Then, the functional Journal of Function Spaces satisfies for any δ > 0 the estimate where Λ 3 and Λ 4 are two positive constants.
Proof. First, we begin to estimate ψ 1 ′ ðtÞ Then, we have As in previous proof and by using Young's inequality, we conclude that for any δ > 0, Similarly and by using the fact kdivuk 2 ≤ ck∇uk 2 , we have The same argument for where Λ 3 = ½ðð2ðp + 1ÞÞ/ðp − 1ÞÞEð0Þ p−1 .

Lemma 7.
Let ðu, v, z, yÞ be a solution of the problem (19). Then, the functional satisfies the estimate Proof. Differentiating (66) with respect to t, we get By using (17) and (18), we have Journal of Function Spaces Thus, Since e −ρ is deceasing function over ðτ 1 , τ 2 Þ, the desired estimate (67) follows immediately from (25).
The following lemmas are needed to prove the general decay when the functions G i ðtÞði = 1, 2Þ are nonlinear. The proof can be found in Mustafa [19].

Lemma 8. The functional
where σ 1 ðtÞ = Ð ∞ t g 1 ðsÞds satisfies Lemma 9. The functional where σ 2 ðtÞ = Ð ∞ t g 2 ðsÞds satisfies Now, we define the following functional where N, N 1 , and N 2 are positive constants. It is easy to prove FðtÞ and EðtÞ are equivalent, namely, there exist two positive constants κ 1 and κ 2 such that By Young's inequality, we get Then, for any N, there exists κ 1 > 0 such that On the other hand, we can find We choose N large enough so that Proof. Let From Lemmas 5, 6, and 7, noting that g i ′ = ζg i − h i we have for any t ≥ t 1 , where Taking δ = 1/2N 2 , we can get First, we take N 1 > 0 large such that Journal of Function Spaces We choose N 2 > 0 large enough so that Note that Then, for any s ∈ ½0,∞Þ, we get Thus, there exist some ζ 0 ð0 < ζ 0 < 1Þ such that if ζ < ζ 0 , then ζC ζ,2 < Thus, (82) is established.