WELL-POSEDNESS AND ASYMPTOTIC STABILITY FOR THE LAM´E SYSTEM WITH INFINITE MEMORIES IN A BOUNDED DOMAIN

. In this work, we consider the Lam´e system in 3-dimension bounded domain with inﬁnite memories. We prove, under some appropriate assump- tions, that this system is well-posed and stable, and we get a general and precise estimate on the convergence of solutions to zero at inﬁnity in terms of the growth of the inﬁnite memories.

with initial conditions u (x, −t) = u 0 (x, t), in Ω, where = ∂ ∂t and u 0 and u 1 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 452 AHMED BCHATNIA AND AISSA GUESMIA and λ and µ are the Lamé constants which satisfy the conditions Moreover, where g i : R + → R + are given functions which represent the dissipative terms.
The problem of well-posedness and stability and/or the obtention of bounded estimates for elasticity systems in general, and the Lamé system in particular, has attracted considerable attention in recent years, where diverse types of dissipative mechanisms have been introduced and several stability and boundedness results have been obtained. The main problem concerning the stability and/or boundedness of estimates of solutions in the presence of finite or infinite memory is to determine the largest class of memory functions which guarantees the stability and/or boundedness of estimates for the system, and the best estimate on the decay rate and/or the bound for solutions in terms of the memory function. Let us recall here some known results in this direction related to our goals, addressing problems of existence, uniqueness and asymptotic behavior of solutions.
1. Damping controls. Real progress has been realized during the last three decades, in particular, in the works of Lagnese [19,20], Komornik [18], Martinez [21], Aassila [1], Alabau and Komornik [2], Horn [14,15], Guesmia [8,9], and Bchatnia and Daoulatli [4]. In [19], Lagnese proved some uniform stability results of elasticity systems with linear feedback and under some technical assumptions on the elasticity tensor. In particular, these results do not hold in the linear homogeneous isotropic case for which the elasticity tensor depends on two parameters called Lamé constants. In [20], Lagnese obtained uniform stability estimates for linear homogeneous isotropic and bidimensional elasticity systems under a linear boundary feedback. Komornik [18] proved the same estimates for the homogeneous isotropic system in 1-dimension and 2-dimension and under a linear boundary feedback. The estimates of Komornik [18] are even optimal when the domain is a ball from R 3 . Martinez [21] generalized the results of Komornik [18] to the case of elasticity systems of cubic crystals under a nonlinear boundary feedback. For these systems, the elasticity tensor depends on three parameters.
Aassila [1] proved the strong stability of a homogeneous isotropic elasticity system with an internal nonlinear feedback in domains of finite Lebesgue measure, but no stability estimate on the decay rate of solutions was given. Alabau and Komornik [2] studied an anisotropic elasticity system with constant coefficients and linear boundary feedback. Under certain geometric conditions, they obtained some exact controllability and uniform stability results, where the decay rate of solutions is given explicitly in terms of the parameters of the system. The proof of [2] is based on the multipliers method and some new identities. Horn [14,15] obtained some stability results for homogeneous isotropic elasticity systems under weaker geometric conditions. The key of the proof in [14,15] is a combination of the multipliers method and the microlocal analysis. Guesmia [8,9] considered the problem of observability, exact controllability and stability of general elasticity systems with variable coefficients depending on both time and space variables in bounded domains or of a finite Lebesgue measure.
The results of [8,9] hold under linear or nonlinear, global or local feedbacks, and they generalize and improve, in some cases, the decay rate obtained by Alabau and Komornik [2]. Recently, Bchatnia and Daoulatli [4] considered the case of the Lamé system in a three-dimensional bounded domain with local nonlinear damping and external force, and obtained several boundedness and stability estimates depending on the growth of the damping and the external forces. The control region considered in [4] satisfies the famous geometric optical condition (GOC).
For the stability of other kind of coupled hyperbolic systems, let us mention the following results. Guesmia [7] considered a coupled wave-Petrovsky system with two nonlinear internal dampings, and showed some polynomial and exponential stability estimates. Alabau, Cannarsa and Komornik [3] considered a coupled system of two abstract hyperbolic equations with linear weak coupling of order zero and only one damping acting on the first equation, and proved that this system is not exponentially stable and the asymptotic behavior of solutions is at least of polynomial type with decay rates depending on the smoothness of initial data. The method introduced and developed in [3] is based on a general estimate on the asymptotic behavior of solutions in terms of higher order initial energies. Some extensions of the results of [3] to the nonlinear and nondissipative cases are given by Guesmia [10] in the particular case of coupled wave equations. Recently, the stability of a coupled Euler-Bernoulli and wave equations with linear weak coupling and clamped boundary conditions for the Euler-Bernoulli equation was considered in Tebou [23]. The decay estimates obtained in [23] are of polynomial type with decay rates smaller than the ones obtained in [3], but the abstract framework introduced in [3] does not include the case considered in [23]. See also the references of [3,10,16,23] for further results related to the stability of coupled hyperbolic equations.
2. Memory controls. The asymptotic stability with finite or infinite memories of hyperbolic partial differential equations has been the subject of many works in the last few years. Let us mention here some works in this direction.
In the case where the memory function converges exponentially to zero, it was proved that the system is exponentially stable; that is, the solution converges exponentially to zero (see [6] and the references therein for abstract dissipative systems). When g does not converge exponentially to zero at infinity, the stability of such systems has been proved in [11], where general decay estimates depending on the growth of the memory function at infinity were obtained. The approach of [11] was applied in [12,13] to, respectively, the wave equation and different kinds of Timoshenko systems. See [11,12,13] for more known results in the literature concerning the stability with finite or infinite memory.
In all stability results with memory cited above, the coupling terms are not a part of the principal operator but additional terms in the system. Concerning the Lamé system with infinite memories (1)-(2) considered in this work, the unique coupling is given in the principal operator, and as far as we know, there is no stability and/or boundedness results in the literature. Our aim in this work is to prove that the stability and/or boundedness of our system holds with infinite memories and to obtain a general decay connection (exponential, polynomial, or others) between the decay rates of the solutions and the growth of the memory functions.
The paper is organized as follows: in Section 2, we prove the global existence and uniqueness of solutions of (1)- (2). Section 3 is devoted to state the main results of this work, that is, the stability of the system (1)- (2). Finally, in Section 4, we prove the stability results.
2. Well-posedness. In this section, we prove the existence and uniqueness of solutions of (1)-(2) using semigroup theory. We consider the following hypothesis: (H1) The functions g i are nonnegative, differentiable and nonincreasing such that Following the idea of [5], we consider Consequently, we obtain where η t = ∂η ∂t and η s = ∂η ∂s . By combining (1) and (5), we obtain the following equation: where The set L g is a Hilbert space endowed with the inner product, Thanks to (3) and (4), the set H is also a Hilbert space endowed with the inner product defined, Now, let U = (u, u , η) T and U 0 = (u 0 (·, 0), u 1 , η 0 ) T . Thanks to (7) and (8), (1)-(2) is equivalent to the abstract linear first-order Cauchy problem where A is the linear operator defined by since g i is nonincreasing. This implies that A is dissipative. On the other hand, we prove that Id − A is surjective; that is, for any W = (w 1 , w 2 , w 3 ) ∈ H, there exists The first and last equations of (11) are equivalent to and 456 AHMED BCHATNIA AND AISSA GUESMIA By integrating the equation (13) with respect to s and noting that v 3 (0) = 0, we obtain Using (12) and (14), the second equation of (11) becomes It is sufficient to prove that (15) has a solution v 1 in H 2 (Ω) ∩ H 1 0 (Ω) 3 , and then we replace in (12) and (14) to conclude that (11) has a solution V ∈ D(A). So we multiply (15) by a test function ϕ 1 ∈ H 1 0 (Ω) 3 and we integrate by parts, obtaining the following variational formulation of (15): where It is clear that a is a bilinear and continuous form on H 1 0 (Ω) 3 × H 1 0 (Ω) 3 , and l is a linear and continuous form on H 1 0 (Ω) 3 . On the other hand, (3) and (4) imply that there exists a positive constant a 0 such that which implies that a is coercive. Therefore, using the Lax-Milgram Theorem, we conclude that (16) has a unique solution v 1 in H 1 0 (Ω) 3 . By classical regularity arguments, we conclude that the solution v 1 of (16) belongs into H 2 (Ω) ∩ H 1 0 (Ω) 3 and satisfies (15). Consequently, using (12) and (14), we deduce that (11) has a unique solution V ∈ D(A). This proves that Id − A is surjective. Finally, using the Lummer-Phillips Theorem (see [22]), we find that A is an infinitesimal generator of a linear C 0 −semigroup on H. Consequently, applying semigroup theory to (9) (see [17,22]), we get the following well-posedness results of (1)-(2): Theorem 2.1. Assume that (3) and (H1) are satisfied. Then, for any the system (1)-(2) has a unique weak solution Moreover, if (u 0 (·, 0), u 1 ) ∈ H 2 (Ω) ∩ H 1 0 (Ω) 3 × H 1 0 (Ω) 3 , the solution of (1)- (2) is classical; that is 3. Stability. In this section, we state our stability results for problem (1)- (2). For this purpose, we start with the following hypotheses: (H2) (H3) For any i = 1, 2, 3, 18) or there exists an increasing strictly convex function G : Remark 1. The condition (19) introduced in [11] is satisfied by any positive function g i of class C 1 (R + ) with g i < 0 and g i is integrable on R + (see [11,12,13] for explicit examples).
The classical energy of any weak solution u of (1)-(2) at time t is defined by We define the "modified" energy functional of the weak solution u by Now, we give our main stability results.
4. Proof of Theorem 3.1. First, we prove Lemmas 4.1-4.3 for classical solutions and we note that these results remain valid for any weak solution by simple density arguments. These Lemmas are well known in the case of the wave equation or Timoshenko systems, see, for example, [11,12,13] and the references therein. On the other hand, we can assume that E(t) > 0, for any t ∈ R + , without loss of generality. Otherwise, if E(t 0 ) = 0, for some t 0 ∈ R + , then E(t) = 0, for all t ≥ t 0 , because E is positive and nonincreasing, and then (23) and (24) are satisfied.