Well-posedness and stabilization of a type three layer beam system with Gurtin-Pipkin’s thermal law

: The goal of this work is to study the well-posedness and the asymptotic behavior of solutions of a triple beam system commonly known as the Rao-Nakra beam model. We consider the e ff ect of Gurtin-Pipkin’s thermal law on the outer layers of the beam system. Using standard semi-group theory for linear operators and the multiplier method, we establish the existence and uniqueness of weak global solution, as well as a stability result.


Introduction
In the present work, we consider the Rao-Nakra (three layer) beam system, where the top and the bottom layers of the beam are subjected to Gurtin-Pipkin's thermal law, namely with the following boundary conditions: u x (0, t) = v x (0, t) = w(0, t) = w xx (0, t) = θ(0, t) = ϑ(0, t) = 0, t ≥ 0, u(π, t) = v(π, t) = w(π, t) = w xx (π, t) = θ x (π, t) = ϑ x (π, t) = 0, t ≥ 0, (1.2) and the initial data u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), w(x, 0) = w 0 (x), x ∈ (0, π), u t (x, 0) = u 1 (x), v t (x, 0) = v 1 (x), w t (x, 0) = w 1 (x), x ∈ (0, π), θ(x, −t) = θ 0 (x, t), ϑ(x, −t) = ϑ 0 (x, t), x ∈ (0, π), t > 0. (1. 3) The relaxation functions g 1 and g 2 are positive non-increasing functions to be specified later.The stabilization of Rao-Nakra beam systems has gathered much interest from researchers recently, and a great number of results have been established.The Rao-Nakra beam model is a beam system that takes into account the motion of two outer face plates (assumed to be relatively stiff) and a sandwiched compliant inner core layer, see [1][2][3][4][5] for Rao-Nakra, Mead-Markus and multilayer plates or sandwich models.The basic equations of motion of the Rao-Nakra model are derived thanks to the Euler-Bernoulli beam assumptions for the outer face plate layers, the Timoshenko beam assumptions for the sandwich layer and a "no slip" assumption for the motion along the interface.Suppose h( j), j = 1, 2, 3 is the thickness of each layer in the beam of length π, see Figure 1 and h = h(1) + h(2) + h(3) the total thickness of the beam.Assuming the Kirchhoff hypothesis is imposed on the outer layers of beam and in addition, there is a continuous, piecewise linear displacements through the cross-sections, Liu et al. [6] gave a detailed derivation of following laminated beam system: where x ∈ (0, π), t > 0, (u, y 1 ), (v, y 3 ) represent longitudinal displacement and shear angle of the bottom and top layers plates.The transverse displacement of the beam is represented by w, and τ is the shear stress of the core layer.Also, for j = 1, 2, 3 (from bottom to top layer), E j , G j , I j , ρ j > 0 are Young's modulus, shear modulus, moments of inertia and density respectively for each layer.Moreover, in (1.4) 3 , we have that ρh = ρ 1 h 1 + ρ 2 h 2 + ρ 3 h 3 and EI = E 1 I 1 + E 3 I 3 .By neglecting the rotary inertia in top and bottom layers of the beam, we obtain and (1.4) 5 .Furthermore, if we neglect the transverse shear, this leads to the Euler-Bernoulli hypothesis w x + y 1 = w x + y 3 = 0. Assuming that the core layer consists of a material that is linearly elastic with the stress-strain relationship τ = 2G 2 ε, where the shear strain ε is defined by Thus, we arrive at the following Rao-Nakra beam model [1], given by where is the Poisson ratio.Furthermore, when the extensional motion of the outer layers is neglected, system (1.4) takes the form of the two-layer laminated beam system derived by Hansen and Spies [7].Li et al. [8] showed that system (1.5) is unstable if only one of the equations is damped.When two of the three equations in (1.5) were damped, the authors in [8] proved a polynomial stability.For recent results in literature, Méndez et al. [9] considered (1.5) with with Kelvin-Voigt damping and studied the well-posedness, lack of exponential decay and polynomial decay.Feng and Özer [10] looked at a nonlinearly damped Rao-Nakra beam system and established the global attractor with finite fractal dimension.Feng et al. [11] studied the stability of Rao-Nakra sandwich beam with time-varying weight and time-varying delay.Mukiawa et al. [12] considered (1.5) with viscoelastic damping on the first equation and heat conduction govern by Fourier's law and proved the well-posedness and a general decay result.Also, Raposo et al. [13] coupled (1.5) with Maxwell-Cattaneo heat conduction established the well-posedness.For more results related Rao-Nakra beam system with frictional, delay or thermal damping, see [14][15][16][17][18][19][20] and the references therein.An interesting tool used by Mathematician in stabilizing beam models such as the Laminated and Timoshenko beam systems is the Gurtin-Pipkin's thermal law, see [21], with constitutive equation where θ = θ(x, t) is the temperature difference, q = q(x, t) is the heat flux, β is a coupling constant coefficient and the relaxation g is a summable convex L 1 ([0, +∞)) function with unit mass.For results related to (1.6), Dell'Oro and Pata [22] studied and proved an exponential stability result if and only if χ h = 0, where For similar results with Gurtin-Pipkin's thermal law, see [23][24][25][26][27][28] and references therein.As clearly elaborated in [22], the Fourier's and Cattaneo's (second sound) thermal law can be recovered from (1.6) by defining the memory function g in (1.6) as respectively.A closely related thermal law to the Gurtin-Pipkin's thermal law is the Coleman-Gurtin's heat conduction law, see [29], with constitutive equation given by where η = 1 and η = 0 correspond to the Gurtin-Pipkin's and Fourier thermal laws, respectively.This entails replacing (1.7) 3 with We should note here that systems govern by Coleman-Gurtin's thermal lawa (1.10) gain additional dissipation from the term − (1−η) β θ xx and thus less difficult to handle compare to systems with Gurtin-Pipkin's thermal law (1.6).
Our main focus of this paper is to investigate the well-posedness and the asymptotic behavior of solutions of system (1.1)- (1.3).We mote here that, the rotational inertia term w xxtt which should be in (1.1) 3 of the original models is neglected in the present model.However, the result in this paper is not affected by the absent of this term.Also, since the thermal coupling in system (1.1)-(1.3) is not strong enough to achieve exponential stability, a viscous damping term w t is added to (1.1) 3 .The rest of work is organized as follows: In Section 2, we state some assumptions and set up our problem (1.1)-(1.3) in appropriate spaces.In Section 3, we prove the existence and uniqueness result for the system (1.1)- (1.3).In Section 4, we study the asymptotic behavior of solution of system (1.1)-(1.3).

Problem transformation
Due to the work of Dafermos [30], we define new functions for the relative past history of θ and ϑ as follows: On account of the boundary conditions (1.2), we have and routine calculation gives where σ 0 and ζ 0 represent the history of θ and ϑ respectively.Also, using direct computations, we have (2.8) Similarly, we get On account of (2.6)-(2.9),system (1.1)-(1.3)takes the form with the boundary conditions and the initial data x ∈ (0, π), s > 0. (2.12) Setting Ψ = (u, φ, v, ψ, w, ϕ, θ, σ, ϑ, ζ) T , with φ = u t , ψ = v t and ϕ = w t .Then, the semi-group formulation of system (2.10)-(2.12) is given by the Cauchy problem (P) where T and the linear operator A is defined by

Functional spaces
Let ⟨, ⟩ and ∥.∥ denote the inner product and the norm in L 2 (0, π) respectively and we define following Sobolev spaces: where H 2 * is equip with the inner product It is easy to check that (H 2 * , ∥.∥ 2 ) is a Banach space and the norm ∥.∥ 2 is equivalent to the usual norm in H 2 (0, π).Next, we introduce the weighted-Hilbert space of H 1 a (0, π)-real valued functions on (0, +∞) by where and equip them with the inner product and norm Also, we define Now, we introduce the phase space of our problem given by and equipped it with the inner product and norm The domain of the linear operator A in (2.13) is defined as follows: (1) Due to (2.5), we can deduce that (2.14) (2) Using Hölder's and Young's inequalities, we have that (2.15)

Well-posedness
In this section, we establish the existence and uniqueness of global weak solution to the system (2.10)-(2.12).

Needed lemmas for well-posedness
Lemma 3.1.The linear operator A : D(A) ⊂ H → H defined in (2.13) is monotone.
From (2.5) and (2.6), we obtain ) ∈ H, we look for a unique solution such that Ψ solves the stationary problem (3. where Now, we observe that last terms in f 4 and f 5 are in H −1 (0, π).Indeed, since k 8 ∈ L 2 µ 1 , we have for any In the same way, we get that Next, we consider the Banach space and equip it with the norm On the account of the weak formulation of (3.5), we consider the bilinear form B on H × H and linear form L on H, define as follows: for every (u, v, w, θ, ϑ), (u * , v * , w * , θ * , ϑ * ) ∈ H. Routine computations, using Cauchy-Schwarz, Young's and Poincaré's inequalities shows that B is a bounded and coercive bilinear form on H × H, and L is a bounded linear form on H. Therefore, using Lax-Milgram theorem, there exists a unique (u, v, w, θ, ϑ) ∈ H such that Then, using standard regularity theory, it follows from (3.5), that Since u, v ∈ H 1 b , w, k 6 ∈ H 2 * and k 6 ∈ L 2 , it easy to see from (3.5) 3 that w satisfy Also, from (3.3), substituting θ and ϑ, we see that Finally, from (3.2) 7 and (3.2) 9 , using regularity theory, we get that Thus, Ψ = (u, φ, v, ψ, w, ϕ, θ, σ, ϑ, ζ) ∈ D(A) and satisfies (3.1).That is, the operator A is maximal.□

Main stability result
The main stability result of this work is the following: holds, then the energy functional E(t) defined in (4.1) decays exponentially.That is, there exists positive constants M and λ such that

Conclusions
In this work, we investigated the the effect of Gurtin-Pipkin's thermal law on the outer layers of the Rao-Nakra beam model.Using standard semi-group theory for linear operators and the multiplier method, the well-posedness and a stability result of solutions of the triple beam system have been established.

Use of AI tools declaration
The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.