Global existence and exponential stability of coupled Lamé system with distributed delay and source term without memory term

In this paper, we prove the global existence and exponential energy decay results of a coupled Lamé system with distributed time delay, nonlinear source term, and without memory term by using the Faedo–Galerkin method. In addition, an appropriate Lyapunov functional, more general relaxation functions, and some properties of convex functions are considered.

After several authors have studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities and is proven under some given conditions (see, for example, [1][2][3][4][5][6][7][8][9][10][11]). In recent years, several authors have been interested in studying the existence and stability for Lamé systems, we refer to [12][13][14]. The Lamé system with localized nonlinear damping and a general decay result of energy have been considered by some recent works (see, for example, [12,14], and [15]). Bchatnia et al. in [16] investigated Lamé systems with past history. Then, Taouaf et al. in [17] established the well-posedness and asymptotic stability for the Lamé system with internal distributed delay.
Beniani et al. [13] proved the well-posedness and exponential stability of the following coupled Lamé system: After that, Baowei et al. in [18] considered the same problem with more general assumption on the relaxation functions. They established an explicit and general decay result, which are optimal, to the system. Boulaaras et al. in [14] considered the previous problem with a source term, where under some suitable conditions on the initial data and the relaxation functions, they proved an asymptotic stability result of global solution taking into account that the kernel is not necessarily decreasing.
In the present work, we prove the existence and general decay results of problem (1.1) with respect to the presence of distributed term delay in order to ensure fast stability under some given conditions. We establish the exponential energy decay results to the system by using an appropriate Lyapunov functional. This paper is organized as follows. In the second section, we give some preliminaries related to problem (1.1). In Sect. 3, we prove the global existence by using Faedo-Galerkin method. In the fourth section, we prove our main result of exponential energy decay.

Preliminaries
In this section, we provide some materials and necessary assumptions which we need in the proof of our results. We use the standard Lebesgue and Sobolev spaces with their scaler products and norms. For simplicity, we would write · instead of · 2 .
(A1) For the source terms f 1 and f 2 , we take with α, β > 0. Clearly, and So, we have the embedding Let c s be the same embedding constant, so we have ν q ≤ c s ∇ν 2 , ν q ≤ c s ν r for ν ∈ H 1 0 ( ). (2.5) As in many papers, we introduce the following new variables: and (2.7) Consequently, problem (1.1) is equivalent to with the initial data and boundary conditions The energy associated with problem (2.8) is defined by (2.10) First, we prove in the following theorem, the result of energy identity.
The first estimate.
The second estimate: First, we are going to estimate u k tt (0) and v k tt (0). Testing the first and second equations in (3.4) by g j,k (t) and h j,k (t) respectively and taking t = 0, we obtain In order to calculate the second estimate, we take the derivatives of the first and second equations of system (3.4) with respect to t, we get u k ttt , w j + μ ∇u k t , ∇w j + (λ + μ) div u k t , div w j Multiplying by (g j,k (t)) and (h j,k (t)) respectively and summing with respect to j from 1 to k, we obtain and (3.31) Differentiating the third and fourth equations in (3.4) with respect to t, we get Multiplying by |μ 1 ( )|c jk and |μ 2 ( )| d jk respectively, integrating over (0, 1) × (τ 1 , τ 2 ), and summing over j from 1 to k, it follows that then we obtain Taking the sum of (3.30), (3.31),(3.32), and (3.33), we get Using Cauchy-Schwarz and Young's inequalities, we conclude For the source term After simplification, we obtain

Corollary 3.3
The sequences of approximate solutions {u k , u k } satisfy, as k → ∞,

(3.48)
Proof The proof is similar to that of [11].
We can complete the proof of theorem as in [2].

General decay
In this section we prove that the solution of problem (2.8)-(2.9) decays generally to a trivial solution using the energy method and a suitable Lyapunov functional.
In the following, we present our main stability result.
satisfies the estimate
We choose N > 0 large enough so that The use of F ∼ E gives E(t) ≤ βe -αt , ∀t ≥ 0.