On the stability of a laminated beam with structural damping and Gurtin-Pipkin thermal law

In this paper, we investigate the stabilization of a one-dimensional thermoelastic laminated beam with structural damping, coupled to a heat equation modeling an expectedly dissipative effect through heat conduction governed by Gurtin-Pipkin thermal law. Under some assumptions on the relaxation function g, we establish the wellposedness for the problem. Furthermore, we prove the exponential stability and lack of exponential stability for the problem. To achieve our goals, we make use of the semigroup method, the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem.


Introduction
In this paper, we investigate the well-posedness and asymptotic stability of a thermoelastic laminated beam with structural damping and Gurtin-Pipkin thermal law, i.e., for (x, t) ∈ (0, 1) × (0, +∞), with the initial and boundary conditions ϕ(x, 0) = ϕ 0 (x), ψ(x, 0) = ψ 0 (x), x ∈ [0, 1], w(x, 0) = w 0 (x), θ(x, 0) = θ 0 (x), x ∈ [0, 1], (2 1 ) ϕ t (x, 0) = ϕ 1 (x), ψ t (x, 0) = ψ 1 (x), x ∈ [0, 1], w t (x, 0) = w 1 (x), θ(−s)| s>0 = θ 0 (s), x ∈ [0, 1], ϕ x (0, t) = ψ(0, t) = w(0, t) = θ x (0, t) = 0, t ∈ [0, +∞), where the functions ϕ(x, t), ψ(x, t), 3w(x, t) − ψ(x, t), θ(x, t), g(s) denote the transverse displacement of the beam, which departs from its equilibrium position, rotation angle, effective rotation angle, relative temperature, and the memory kernel, respectively; w(x, t) is proportional to the amount of slip along the interface at time t and longitudinal spatial variable x; g(s) is the heat conductivity relaxation kernel, whose properties will be specified later; (1) 3 describes the dynamics of the slip; ρ, G, I ρ , D, γ, β are the density of the beams, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness of the beams, and adhesive damping parameter, respectively. Moreover, ρ, G, I ρ , D, δ, γ, α, k, β are positive constant. Problem (1) is closely related to 1D thermoelastic Timoshenko beam model in the sense that (1) reduces to the Timoshenko system with Gurtin-Pipkin thermal law [12] studied by Dell'Oro and Pata [9] if the slip w is assumed to be identically zero. When there is no thermal effect, problem (1) is called laminated beam. Hansen [13] derived a model for a two-layered plate in which slip could occur along the interface. Concerned with the beam analog, with strain-rate damping as in the above described plate model [13,Eq. (3.16)], the basic evolution equations for the system are given by where S is the shear force, and M is the bending moment. The constitutive equations are S = G(ψ − ϕ x ), M = D(3w − ψ) x . Hansen and Spies [14] derived the mathematical model for two-layered beams with structural damping due to the interfacial slip, namely, for (x, t) ∈ (0, 1) × (0, +∞). Later on, Wang et al. [29] considered system (3) with the cantilever boundary conditions and two different wave speeds ( G/ρ and D/I ρ ), they pointed out that system (3) can reach the asymptotic stability, but it does not reach the exponential stability due to the action of the slip w. To achieve the exponential decay result, the authors in [29] added an additional boundary control such that the boundary conditions become where ξ = 3w − ψ, and k 1 and k 2 are positive constant feedback gains. Furthermore, Cao et al. [3] proved the exponential stability for system (3) with following boundary conditions: provided k 1 = ρ/G and k 2 = I ρ /D. More importantly, the authors proved that the dominant part of the system is itself exponentially stable. Concerning a laminated beam with thermoelastic dissipation effective in the bending moment, we have where θ is the temperature difference, q denotes the heat flux, S = G(ψ − ϕ x ), and M = D(3w − ψ) x − δθ. Derivative of the heat flux term in the formulation of the rate equation was introduced independently by Cattaneo [4] and Vernotte [28] with a fixed constant κ > 0 and small τ > 0. Combining (4) and (5), Apalara [1] considered a laminated beam with structural damping and second sound for (x, t) ∈ (0, 1) × (0, +∞). The stabilization of system (6) has been analyzed in [1], where Apalara obtained the well-posedness and uniform stability results depending on the following stability number: http://www.journals.vu.lt/nonlinear-analysis Mukiawa et al. [23] studied a thermoelastic laminated beam system without structural damping, but with a finite memory acting on the bending moment and established a general and optimal decay estimate. If we assume Gurtin-Pipkin thermal law [12] of heat conduction where g is called the memory kernel, we can obtain equation (1) 4 . The aim of this paper is to study the well-posedness and asymptotic stability of a thermoelastic laminated beam with structural damping and Gurtin-Pipkin thermal law, i.e., (1)- (2). In fact, Cattaneo law (5) can be reduced as a particular instance of (7), which have been proved in [9]. For other asymptotic behavior results to laminated beams, we refer the reader to [6,14,16,21,29] and the references therein.
In this paper, we first prove the well-posedness by using Lumer-Phillips theorem. And then, by using the perturbed energy method, we establish an exponential stability result depending on the stability number To overcome the difficulty brought by Gurtin-Pipkin thermal law, we use some appropriated multipliers to construct a Lyapunov functional. For the case χ g = 0, we prove the lack of exponential stability by using Gearhart-Herbst-Prüss-Huang theorem. The remaining part of this paper is organized as follows. In Section 2, we introduce some hypotheses and present our main results. In Section 3, we prove the well-posedness for problem (1)- (2). In Section 4, we establish an exponential decay result to problem (1)- (2). In Section 5, we prove the lack of exponential stability for problem (1)-(2). Section 6 is devoted to the conclusion and open problem. Throughout this paper, we use c to denote a generic positive constant.

Preliminaries and main results
In this section, we first introduce some notation and present our hypotheses. Then we give some lemmas, which will be used in the proof of main results.

change of variable and a formal integration by parts yield
Hence system (1)-(2) can be written as http://www.journals.vu.lt/nonlinear-analysis for (x, t) ∈ (0, 1) × (0, +∞) with initial and boundary conditions For the memory kernel g, we assume g ∈ C 2 (R + ) ∩ W 1,1 (R + ) and (G1) g is a bounded convex summable function on [0, ∞); (G2) g has a total mass ∞ 0 g(s) ds = 1; (G3) g is an absolutely continuous function on R + so that (G4) There exists a positive constant ξ so that, for almost every s > 0, Remark 1. In particular, µ is summable on R + with ∞ 0 µ(s) ds = g(0). Furthermore, noting that g(s) has total mass 1, we have ∞ 0 sµ(s) ds = 1. Next, we introduce the vector function U = (ϕ, u, 3w − ψ, 3v − u, w, v, θ, η) T with u = ϕ t and v = w t . Then system (8)-(9) can be written as where A is a linear operator defined by We consider the following spaces: and the energy space Besides, H is the Hilbert space equipped with the norm and the inner product The domain of A is given by The energy associated with problem (8)-(9) is defined by Now, we give our main results in this paper as follows.
Based on two propositions from [9,Props. 11,12], we give the full equivalence between Cattaneo law and Gurtin-Pipkin thermal law.  To obtain the well-posedness, we need to prove that A : D(A) → H is a maximal monotone operator. To achieve this goal, we need to prove that A is dissipative and Id − A is surjective.
As a consequence, A is a maximal monotone operator. Therefore, we established the well-posedness result stated in Theorem 1 by using Lumer-Phillips theorem (see [2]).
Proof. By differentiating F 4 (t) with respect to t, using (8) 1 and integrating by parts, we obtain Using Young's and Poincaré's inequalities, we obtain Then estimate (26) is obtained.
Lemma 7. Let (ϕ, ψ, w, θ) be the solution of (8)- (9). Then the functional F 6 (t) = I ρ 1 0 ww t dx satisfies the estimate Proof. By differentiating F 6 (t) with respect to t, using (8) 3 and integrating by parts, then use Young's inequality to obtain (28). This completes the proof. Now we define the following Lyapunov functional where N , N 1 , N 2 , N 3 are positive constants to be selected later. Then we have the lemma as follows.
Lemma 8. Let (ϕ, ψ, w, θ) be the solution of (8)- (9). For N large enough, there exists a positive c such that, for any t 0, Proof. Using Young's, Poincaré's and Cauchy-Schwarz inequalities, and the fact that (see [22]) we can easily obtain that where α i (i = 1, 2, . . . , 9) are positive constants. It follows from (12) and (29) that there exists a positive constant c such that |L (t) − N E(t)| cE(t), which completes the proof. Now, we are ready to prove the main result in this section.

Conclusion and open problem
In this paper, we first prove the well-posedness for a laminated beam with Gurtin-Pipkin thermal law and structural damping. Then we prove that the system is exponentially stable if and only if that stability number is equal to zero (χ g = 0). When the stability number is not zero (χ g = 0), the problem of whether it is possible to get the polynomial stability for system (8)-(9) is still an interesting open problem. Recently, Guesmia [11] considered the stability of the laminated beam with interfacial slip and infinite memory acting only on the transverse displacement, the rotation angle, and the amount of slip, respectively. He combined the energy method and the frequency domain approach to show that the infinite memory is capable alone to guarantee the strong and polynomial stability of the model, and mentioned also that "when the exponential stability is not satisfied, obtaining the optimal decay rate of solutions is, in our opinion, a very nice and hard question".