Well-posedness and asymptotic stability for the Lamé system with internal distributed delay

(1.2) { u(x) = u0(x), u ′(x, 0) = u1(x), in Ω, u′(x,−t) = f0(x,−t), in Ω× (0, τ2), where (u0, u1, f0) are given history and initial data. Here ∆ denotes the Laplacian operator and ∆e denotes the elasticity operator, which is the 3×3 matrix-valued differential operator defined by ∆eu = μ∆u+ (λ+ μ)∇(div u), u = (u1, u2, u3) and μ and λ are the Lamé constants which satisfy the conditions (1.3) μ > 0, λ+ μ ≥ 0. Moreover, μ2 : [τ1, τ2] → R is a bounded function and τ1 < τ2 are two positive constants.


Introduction
Let Ω be a bounded domain in R 3 with smooth boundary ∂Ω.Let us consider the following Lamé system with a distributed delay term: (1.1) where (u 0 , u 1 , f 0 ) are given history and initial data.Here ∆ denotes the Laplacian operator and ∆ e denotes the elasticity operator, which is the 3 × 3 matrix-valued differential operator defined by ∆ e u = µ∆u + (λ + µ)∇(div u), u = (u 1 , u 2 , u 3 ) T and µ and λ are the Lamé constants which satisfy the conditions Moreover, µ 2 : [τ 1 , τ 2 ] → R is a bounded function and τ 1 < τ 2 are two positive constants.
Lamé system with internal distributed delay In the particular case λ + µ = 0, ∆ e = µ∆ gives a vector Laplacian, that is, (1.1) describes the vector wave equation.
In recent years, the control of partial differential equations with time delay effects has become an active and attractive area of research see ( [1,7,9,14,15,16] and [21]), and the references therein.Recently, S. A. Messaoudi et al. [21] considered the following problem with a strong damping and a strong distributed delay: and under the assumption The authors proved that the solution is exponentially stable.
In [3], the authors considered the Bresse system in bounded domain with internal distributed delay where (x, t) ∈]0, L[×R + , the authors proved, under suitable conditions, that the system is well-posed and its energy converges to zero when time goes to infinity.For Timoshenko-type system with thermoelasticity of second sound, in the presence of a distributed delay Apalara [1] considered the following system: and proved an exponential decay result under the assumption In [4], Beniani and al. considered the following Lamé system with time varing delay term: (1.9) the authors proved, under suitable conditions, that energy converges to zero when time goes to infinity.The paper is organized as follows.In Section 2, we prove the global existence and uniqueness of solutions of (1.1)-(1.2).In Section 3, we prove the stability results.

Well-posedness
In this section, we prove the existence and uniqueness of solutions of (1.1)-(1.2) using semigroup theory.
Next, we will formulate the system (1.1)-(1.2) in the following abstract linear first-order system: We define the inner product in H, The operators A is linear and given by (2.4) The domain D(A) of A is given by The well-posedness of problem (2.3) is ensured by the following theorem.
Theorem 2.1.Assume that Then, for any U 0 ∈ H , the system (2.3) has a unique weak solution Moreover, if U ∈ D(A), then the solution of (2.3) satisfies (classical solution) Proof.We prove that A : D(A) → H is a maximal monotone operator, that is, A is dissipative and Id − A is surjective.Indeed, a simple calculation implies that, for any AV, V H = µ Ω ∇v(x, t)∇u(x, t)dx Using Young's inequality and taking into account that z(., 0, ., .)= v, we get by virtue of (2.5).Therefore, A is dissipative.On the other hand, we prove that Id − A is surjective.Indeed, let F = (f, g, h) T ∈ H we show that there exists Using the equation in (2.9), we obtain z(x, t, ρ, s) = (u − f )e −ρs + e −ρs ρ 0 sh(x, σ)e σs dσ.

Stability
In this section, we investigate the asymptotic behaviour of the solution of problem (2.3).In fact, using the energy method to produce a suitable Lyapunov functional, we define the energy associated with the solution of (1.1)-(1.2) by (3.1) Theorem 3.1.Assume that (1.3) and (2.5) hold.Then, for any U 0 ∈ H, there exist positive constants δ 1 and δ 2 , such that the solution of (2.3) satisfies We carry out the proof of Theorem 3.1.Firstly, we will estimate several Lemmas.Lemma 3.2.Suppose that µ 1 , µ 2 satisfy (2.5).Then energy functional satisfies, along the solution u of (1.1)-(1.2), Using (2.2) and integrating by parts, we get (3.5) Young's inequality leads to the desired estimate.
Lemma 3.4.The functional Proof.Using (2.1), the derivative of I entails (3.12)At this point, we choose our constants in (3.14), carefully, such that all the coefficients in (3.14) will be negative.It suffices to choose so small such that e −τ 2 − c > 0, then pick N large enough such that Consequently, recalling (3.1), we deduce that there exist also η 2 > 0, such that (3.15) dL(t) dt ≤ −η 2 E(t), for t ≥ 0.
On the other hand, it is not hard to see that from (3.13)
and the desired estimate follows immediately.Now, we prove our main stability results (3.2).