EXPONENTIAL STABILITY OF A TIMOSHENKO TYPE THERMOELASTIC SYSTEM WITH GURTIN-PIPKIN THERMAL LAW AND FRICTIONAL DAMPING

In this paper we consider a linear thermoelastic system of Timoshenko type where the heat conduction is given by the linearized law of GurtinPipkin. An existence and uniqueness result is proved by the use of a semigroup approach. We establish an exponential stability result without any assumption on the wave speeds once here we have a fully damped system.


Introduction
In the present paper we investigate the well-posedness and the asymptotic behavior of the following Timoshenko type system where u is the transverse displacement of a beam of length π, φ is the rotation angle of filament, θ is the temperature variation from an equilibrium reference value and q is the heat flux. The coefficients ρ 1 , ρ 2 , c, κ, τ are positive and present the mass density, the polar moment of inertia of a cross section, the specific heat constant, the shear modulus and the intensity of the frictional damping respectively, b = EI is the product of Young's modulus of elasticity and the moment of inertia of a cross section, β and δ are coupling constants that are different from zero but their signs does not matter in the analysis. To render the system (1) determined an additional equation relating q and θ is needed. In the classical theory of thermoelasticity the constitutive equation for the 96 A. FAREH heat flux is expressed through Fourier's law of heat conduction where k > 0 represents the coefficient of the thermal conductivity of the material.

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Alves et al. [2] improve the results of [1] for the case of different wave speeds and obtained the same rate of decay t −1/2 independently of the boundary conditions. Later, Alves et al. [3] extended the results of [1] to the non-homogeneous case with the boundary conditions (7). Precisely, they established an exponential stability provided that the non-homogeneous wave speeds satisfy the condition in an open subinterval I of (0, L). When (10) does not hold they obtained a polynomial stability result with a rate of decay depending on the regularity of the initial data.
Recently, Jorge-Silva and Racke [19] considered (6) with Cattaneo's law and proved that there is non exponential stability no matter if (9) holds which confirms the result of [10].
We recall that the model using Fourier's law (2) leads to a parabolic equation. Consequently, the heat propagates with an infinite speed, that is, any thermal disturbance produced at some point in the body has an instantaneous effect elsewhere in the body. To overcome this physical paradox, many theories were developed. Green and Naghdi [12][13][14] expanded three new theories based on an entropy equality rather than the entropy inequality. They called them thermoelasticity of type I, type II and type III respectively. In each of these theories the equation for the heat flux is given by a different constitutive assumption. The constitutive equation for the heat flux in the type III theory is given by is the thermal displacement and f 1 , f 2 are two positive constants.
In the framework of the thermoelasticity of type III, Messaoudi and Said-Houari [24] considered the following Timoshenko type system and showed that the solution (φ, ψ, θ) decays exponentially provided that K The case of non equal speeds was examined by Messaoudi and Fareh [23]. They established a polynomial rate of decay. Fatori et al. [9] show that the optimal rate in this case is t −1/2 .

A. FAREH
Santos and Almeida Júnior [30] extended the results of [23,24] to the Timoshenko system with thermoelastic effect acting on a shear force The second theory proposed to overcome the paradox of infinite speed was developed by Lord and Shulman [21]. They suggested to replace Fourier's law (2) by Cattaneo's one τ 0 q t + q + kθ x = 0, where the positive constant τ 0 represents the time lag in the response of the heat flux to the temperature gradient and is referred to as the thermal relaxation time. According to this theory, the system becomes fully hyperbolic, as a result the heat propagates with a finite speed and is viewed as a wave-like propagation rather than a diffusion phenomenon. A wave-like thermal disturbance is referred to as a second sound (where the first sound being the usual sound) and a nonclassical theory predicting the occurrence of such disturbances are known as thermoelasticity with finite wave speeds or second sound thermoelasticity.
Fernández Sare and Racke [10] considered the following Timoshenko type system with second sound thermoelasticity and proved that the solution of (11) is no longer exponentially stable even if ρ 1 k = ρ 2 b . However, the incorporation of the frictional damping µφ t into the first equation of (11) produces an exponential stability independently of the wave speeds [22].
Santos et al. [31] introduced the stability number and proved that the solution of (11) is exponentially stable provided that χ 0 = 0. It is worth noting that the type III thermoelasticity and the second sound thermoelasticity are unable to describe the memory effect which reigns in some materials, particularly at a low temperature. This fact leads to the look for a more general constitutive assumption relating the heat flux to the thermal memory. Gurtin and Pipkin [16] assumed that the heat flux depends on the integrated history of the temperature gradient, and established a general nonlinear theory for which thermal disturbances propagate with a finite speed. In accordance with this theory, the linearized constitutive equation for q is given by

EXPONENTIAL STABILITY OF TIMOSHENKO SYSTEM WITH G-P THERMAL LAW 99
where k(s) is the heat conductivity relaxation kernel. The presence of the convolution term (12) renders the Timoshenko system coupled with the heat equation into a fully hyperbolic system, which allows the heat to propagate with a finite speed and admits to describe the memory effect of the heat conduction.
In the context of Gurtin-Pipkin theory Pata and Vuk [26] studied the linear thermoelastic system where the heat flux q is given by (12). They proved, under some assumptions on µ (s) = −k ′ (s), that the solution of the system decays exponentially. Fatori and Muñoz Rivera [8] considered the system and established an exponential decay result provided that the kernel k is positive definite and decays exponentially.
Concerning Timoshenko systems coupled with the heat equation in the framework of Gurtin-Pipkin's theory, Dell'Oro an Pata [7] analyzed the following system and proved that the semigroup associated with the solution of the system (13) is exponentially stable if and only if Closely related to Timoshenko's beam theory, Raposo [29] investigated the lami- and obtained an exponential stability result. Regarding the damping by the heat conduction, Liu and Zhao [20] showed that the laminated beam coupled with the heat equation modelled via Fourier's law of the heat conduction is exponentially stable provided that the wave speeds are equal. Apalara [4] obtained the same result by coupling the laminated beam with the heat equation moddeled via Cattaneo's 100 A. FAREH law, provided that the equal wave speeds is replaced by a relation between the coefficients of the system. Choucha et al. [5] added a distributed delay and proved the exponential and the polynomial stability for the equal and the non-equal wave speeds respectively. They also kept the same results in the presence of a viscoelastic damping and a distributed delay [6].
In view of the aforementioned studies we can summarized the stability results for Timoshenko systems coupled with thermal effects as follows: i) A fully damped Timoshenko system with parabolic thermal effects is exponentially stable regardless any restriction on the wave speeds. ii) A Timoshenko system damped only by thermal effects is exponentially stable if and only if the coefficients of the system satisfy a stability condition (equal wave speeds, in the case of the classical parabolic heat equation).
To the best of my knowledge there is no results concerning the fully damped Timoshenko system with hyperbolic thermal dissipation. One can expected that this leads to an exponential stability. In the present paper we give a positive answer to this concern.
It should be noted here, that replacing the parabolic heat conduction by a hyperbolic type one is not obviously profitable, first, because the system becomes fully hyperbolic and therefore it loses the exponential decay reached with one dissipation when (5) holds, (see [10,28]), secondly, because the dissipative effects due to the hyperbolic type heat conduction are generally weaker than those induced by Fourier's law.
In the present paper we consider the fully damped case of (13) and prove the exponential stability of the solution without any condition. The importance of our result manifested from the fact that the case of equal speeds is purely mathematical, since it is physically never satisfied [15]. Therefore, the stability result obtained without any restriction on the coefficients is more realistic than that obtained with a stability condition.
Note that the presence of the convolution term in the constitutive equation for q renders the family operators mapping the initial value (u 0 , u 1 , φ 0 , φ 1 , θ 0 ) into the solution (u, φ, θ) not match the semigroup properties. This is due to the fact that the solution value of θ at time t depends on the whole function up to time t.
In order to overcome this difficulty we introduce the new variables which denote the past history and the summed past history of θ up to t, respectively. Clearly η t (x, s) satisfies the boundary conditions η(0, s) = η(π, s) = 0.

EXPONENTIAL STABILITY OF TIMOSHENKO SYSTEM WITH G-P THERMAL LAW 101
Moreover, we assume that k (∞) = 0 and η(x, 0) = lim Setting µ (s) = −k ′ (s) , the system (1) and equations (12), (15) The system (16) is complemented with the boundary conditions and the initial data Regarding the memory kernel µ, we assume the following set of hypotheses: ∞ 0 µ (s) ds = k 0 > 0, (h4) there exists ξ > 0, such that µ ′ (s) ≤ −ξµ (s) , ∀s ≥ 0. The rest of the paper is organized as follows: in Section 2, we introduce some functional preliminaries. Section 3 is devoted to the proof of an existence and uniqueness result. In Section 4, we state and prove our stability result.

Functional Setting
Let A = −D 2 be the operator defined over L 2 (0, π) . It is well known that the operator A with the Dirichlet boundary conditions is a self-adjoint and positive operator with domain D (A) = H 2 ∩ H 1 0 . Thus, it is possible to define the powers A α of A for α ∈ R, and the Hilbert space V α = D A α/2 endowed with the inner product ⟨u, v⟩ α = A α/2 u, A α/2 v and the associated norm denoted by ∥u∥ α . In particular, We shall also need to define the spaces M 0 =L 2 µ (0, +∞) ; L 2 (0, π) and The following lemma will be useful in the proof of our main result.
If v ̸ = 0, then □ EXPONENTIAL STABILITY OF TIMOSHENKO SYSTEM WITH G-P THERMAL LAW 103

Well Posedness
In this section we prove that the problem determined by (16)-(18) has a unique solution. The main tools of the proof are the Lumer-Phillips and the Lax-Milgram theorems. First we need to rewrite the problem in the semigroups setting.
Let H be the Hilbert space and the associated norm We note that by virtue of the inequalities the above norm in H is equivalent to the usual norm. Therefore, we use either of the norms indifferently. To rewrite the system (16) in the semigroup setting we introduce the new variables v = u t and ϕ = φ t , then the system (16) becomes and the problem (16)-(18) rewritten
Proof. First, we prove that A is dissipative. Indeed, for every U ∈ D (A) we have

EXPONENTIAL STABILITY OF TIMOSHENKO SYSTEM WITH G-P THERMAL LAW 105
For the second term in the right-hand side, we have Since µ (s) ∥η x (s) ∥ 2 and µ (s) ∥η xs (s) ∥ 2 belong to L 1 (R + ) and η x (0) = 0, hence The Cauchy-Schwarz inequality, leads to The left-hand side of the last equation is bounded, and from (h2) both terms on the right-hand side are positive. Then, the limit in the right hand side exists and is finite, and therefore equals zero. Thus, which proves the dissipativeness of A. Next, we show that A is maximal. Let U * = (u * , v * , φ * , ϕ * , θ * , η * ) T ∈ H, and find U = (u, v, φ, ϕ, θ, η)

EXPONENTIAL STABILITY OF TIMOSHENKO SYSTEM WITH G-P THERMAL LAW 107
and Clearly, B (·, ·) is a bounded bilinear form over W = H 1 0 × H 1 * × H 1 0 and L is a bounded linear form. Furthermore, we have Therefore, there exists a positive constant α such that Thus, B (·, ·) is coercive and by means of the Lax-Milgram theorem, the problem (30) has a unique solution (u, φ, θ) ∈ W.

Asymptotic Behavior
In this section we establish an exponential rate of decay for the solution of the system (16)- (18). The following Lemma gives a sufficient condition for a C 0 −semigroup in order to be exponentially stable.

Lemma 2.
[11] Let S(t) be a contraction semigroup on H, and let A be its infinitesimal generator. If the operator iβI − A is bounded below as β ∈ R, that is there exists λ > 0 such that The main result of this paper reads as follows:

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Theorem 4. Assume that the memory kernel µ satisfies the hypotheses (h1)-(h5). Then the semigroup S(t) = e At associated to the problem (16)-(18) is exponentially stable.
Proof. The proof will be done by a contradiction argument. Suppose that the assertion is false. Then there exist a sequence (λ n ) ⊂ R and a sequence (U n ) ⊂ D (A) , of unit norm which reads in components as iλ n η n − θ n + D s η n −→ 0 in M 1 .
(37) Note that since the norm in H is equivalent to the usual norm, then there exists γ > 0 such that for any U ∈ D (A) of unit norm, we have (38) First we have Thus, ∥ϕ n ∥ −→ 0 (39) and Moreover, from (34) we have The injection L 2 → H −1 is continuous, hence (33) holds in H −1 instead of L 2 and iρ 1 λ n v n ∼ κD 2 u n + κDφ n − βDθ n in H −1 .
At this point we distinguish two cases. Suppose that (λ n ) is unbounded, then we can choose a subsequence (λ n ) such that |λ n | −→ ∞ and from (42) we have v n −→ 0 in H −1 (0, π).
From the uniqueness of the limit we conclude that v = 0, which is incompatible with (54). Conversely, assume that (λ n ) is bounded, again, there exists a subsequence (λ n ) that converges to some λ ∈ R. In this case we have lim n−→∞ ∥(iλI − A) U n ∥ = 0, and (32)-(37) hold with λ instead of λ n . In particular iλu n − v n −→ 0 in H 1 0 (0, π). Since (u n ) is bounded in H 1 0 (0, π), we conclude that there exists v * ∈ H 1 0 (0, π) and a subsequence (v n ) that converges weakly to v * in H 1 0 (0, π). From the uniqueness of the limit we infer that v * = v, which is in contradiction with v * ∈ H 1 0 (0, π), since v is a non-zero constant function, and therefore cannot be in H 1 0 (0, π). This completes the proof of Theorem 4. □

Declaration of Competing Interests
The author declare that he has no conflict of interest.