On a Thermoelastic Laminated Timoshenko Beam: Well Posedness and Stability

In this paper, we are concerned with a linear thermoelastic laminated Timoshenko beam, where the heat conduction is given by Cattaneo’s law. We firstly prove the global well posedness of the system. For stability results, we establish exponential and polynomial stabilities by introducing a stability number χ.


Introduction
In this paper, we address the following thermoelastic laminated Timoshenko beam in (0, 1) × (0, ∞): which subject to the following boundary conditions: , and initial conditions ω(x, 0) � ω 0 (x), ψ(x, 0) � ψ 0 (x), s(x, 0) � s 0 (x), θ(x, 0) � θ 0 (x), x ∈ (0, 1), q(x, 0) � q 0 (x), ω t (x, 0) � ω 1 (x), ψ t (x, 0) � ψ 1 (x), s t (x, 0) � s 1 (x), x ∈ (0, 1), where ρ, G, I ρ , D, c, β, ρ 3 , δ, τ, and α are positive constants. θ(x, t) represents the difference temperature and q(x, t) is the heat flux. [1,2]. ey introduced a mathematical model for two-layered beams with structural damping due to the interfacial slip which is given by where the coefficients ρ, G, I ρ , D, c, and β are positive constants and represent density, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness, and adhesive damping parameter, respectively. e function ω(x, t) denotes the transversal displacement, ψ(x, t) represents the rotational displacement, and s(x, t) is proportional to the amount of slip along the interface at time t and longitudinal spatial variable x. e third equation describes the dynamics of the slip. Up till now, there are some results concerning laminated beam equations, which are mainly concerned with global existence and stability of the related system. By adding suitable damping effects, such as internal damping, (boundary) frictional damping, and viscoelastic damping, it was shown that if the linear damping terms are added in two of the three equations, system (4) is exponentially stable under the "equal wave speeds" assumption (ρ/I ρ ) � (G/D). But if the damping terms are added in the three equations, then the system decays exponentially without the equal wave speeds assumption, see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For thermoelastic laminated Timoshenko beam, there are few published works, we can mention the results due to Liu and Zhao [18] and Apalara [19]. In [18], the authors considered the following laminated beams with past history together with the following boundary conditions: ey firstly proved the global well posedness of solutions to the system. e main results are the stability of the system. If β ≠ 0, they proved the exponential and polynomial stabilities depending on the behavior of the kernel function g only. If β � 0, they established exponential stability in case of equal wave speeds assumption and lack of exponential stability in case of nonequal wave speeds assumption. Apalara [19] considered a laminated beam with second sound of the form together with the following boundary conditions: and proved the global well posedness and established exponential and polynomial stabilities depending on the parameter One can also refer to two recent results of laminated beams with thermal damping in [20,21], and a result of a coupled hyperbolic equations with a heat equation of second sound in [22].
When s � 0, system (4) reduces to the well-known Timoshenko system, which have been widely studied. ere are so many papers on the Timoshenko system in the literature, most of those results recover the global well posedness, stability, and long-time dynamics by adding some kinds of damping. Here, we recall some works on the thermoelastic Timoshenko system. Muñoz Rivera and Racke [23] considered a Timoshenko system with thermoelastic dissipation and established exponential stability in case of equal wave speed assumption and polynomial stability if wave speeds are nonequal. Almeida Júnior et al. [24] studied 2 Complexity a thermoelastic Timoshenko beam acting on shear force. ey obtained the same stability results as in [23]. In addition, they proved that the polynomial decay is optimal. Fernández Sare and Racke [25] considered a Timoshenko system with second sound. ey proved that the system is not exponentially stable even if the propagation speeds are equal. e results were generalized by Guesmia et al. [26]. Recently, Santos et al. [27] introduced a stability number χ r for the system in [25] and established the exponential decay result for χ r � 0 and polynomial decay for χ r ≠ 0 by using the semigroup method. One can also find a stability result for the Timoshenko system with second sound in Apalara et al. [28]. Feng [29] considered a Timoshenko-Coleman-Gurtin system and studied the long-time dynamics of the system. We at last mention the contribution of Hamadouche and Messaoudi [30] and Aouadi and Boulehmi [31], where the authors considered two classes of nonuniform thermoelastic Timoshenko systems and proved global well posedness and established some stability results.
Our goals in the present work are to study the global well posedness and stability of systems (1)-(3). e main points are summarized as follows: and we show that the system is exponential stable when χ � 0 and polynomial stable when χ ≠ 0. e main results are presented in eorems 1 and 2. (iii) e proof of stability results is based on the multiplier method. Since the boundary conditions here we considered are different from those in Apalara [19], so the multipliers we will define are greatly different from the multipliers in Apalara [19].
It follows, from (1), that If we denote we easily verify that (ω, ψ, s, θ, q) satisfies (1) and in addition, Hence, Poincaré's inequality holds for ω. In the following, we work with ω and q but write ω and q for convenience. e remaining paper is planned as follows. In Section 2, we study the well posedness of the system. In Section 3, we establish the stability results. roughout this paper, c > 0 is a generic constant that changes from one inequality to another.

Well Posedness
We start by denoting the vector-valued function by U: Ψ � ψ t , and Λ � s t . (14) en, systems (1)-(3) can be written as d dt where the operator A is defined by We consider the following spaces: Let be the Hilbert space equipped with the inner product e domain of A is given by e well posedness result can be stated in the following theorem.
Proof. It is easy to obtain that, for any U � (ω, Φ, 3s − ψ, which implies the operator A is a dissipative operator.

Stability
In this section, we study the stability of systems (1)-(3). More precisely, we establish exponential and polynomial decay results depending on χ defined by e energy functional of systems (1)-(3) is defined by Now we give our stability results.
Theorem 2 (exponential decay). Suppose that χ � 0. For any initial data U 0 ∈ H, there exist two positive constants μ and η such that the energy functional (48) satisfies Theorem 3 (polynomial decay). Suppose that χ ≠ 0. For any initial data U 0 ∈ D(A), there exists positive constant μ 0 such that the energy functional (48) satisfies To prove eorems 1 and 2, we need the following technical lemmas.

Lemma 2. Define the functional F 1 (t) by
en, we have for any ε 1 > 0, where c * > 0 is the Poincaré constant.

Lemma 3.
e functional F 2 (t) defined by satisfies for any ε 2 > 0, Proof. Differentiating F 2 (t) with respect to t and using (1), we see that (58) Using integration by parts, we obtain (59) en, by using Young's inequality and Hölder's inequality, we can get (57). (60) en, we can get for any ε 3 > 0, Proof. Differentiating F 3 with respect to t and using (1), we obtain Integration by parts gives us By using Young's inequality and Hölder's inequality, we can get (61).

e functional F 4 (t) defined by
satisfies for any ε 4 > 0, Proof. We take the derivative of F 4 and use (1) and integrate by parts to obtain en, using Young's inequality, we can get (65). □ Lemma 6. Define the functional F 5 (t) by en, we have for any ε 5 > 0, where c i (i � 1, 2, 3) are positive constants.
Proof. follows from (1) that Young's inequality gives us (82). Proof. We define the functional L(t) by where N and N i (i � 2, 3, 4, 5) are positive constants that will be chosen later. Note that Replacing (84) in (57) and then combining (51)-(53), (57)-(68), and (82), we obtain Complexity 9 Taking ε 1 � 1, we obtain At this point, we first choose N 5 > 0 large enough such that τGδI ρ 4 For fixed N 5 , we take N 2 > 0 so large that en, we pick N 4 > 0 large so that And then we choose N 3 so large that At last, we take N > 0 large enough so that the functional L(t) is equivalent to the energy functional E(t), i.e., there exist two positive constants: and further so that Recalling (48), we infer that there exists a positive constant β 3 such that, for any t > 0, which, along with (92), implies Integrating (95) over (0, t), we have, for any t > 0, which, using (95) again, gives us (49). e proof of eorem 1 is done. In this section, we consider the case χ ≠ 0 to prove eorem 2.
Differentiating system (1) with respect to time, we obtain the following system: which subject to the following boundary conditions: For any initial data U 0 ∈ D(A), system (97) is well posed. Next, we introduce second-order energy functional E(t) by 10 Complexity By using the same arguments as in Lemma 3, we can get the second-order energy E(t) defined by (99) is nonincreasing and satisfies In Lemma 6, we have proved that, for any ε 5 > 0, anks to (1) and Young's inequality, we derive that (104) Proof. We define the functional L(t) by It follows from (51)-(53), (57)-(65), and (100)-(104) that

Complexity 11
With the same choice of constants as in Section 3.2, we further take N > 0 so large that Noting that (48), we know that there exists a positive constant μ 1 such that, for any t > 0, Since the energy functional E(t) is positive and nonincreasing, we infer (108) that, for any t > 0, which gives us Here, μ 0 � (L(0)t/μ 1 ) � (E(0) + E(0)/μ 1 ). e proof is complete.
□ Remark 1. We point out that the functional L(t) is inequivalent to the energy functional E(t). at is to say, (92) does not hold true.

Data Availability
No data were used during this study.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.