Exponential and polynomial decay for a laminated beam with Fourier ’ s type heat conduction

In this paper, we study the well-posedness and the asymptotic behavior of a onedimensional laminated beam system, where the heat conduction is given by Fourier’s law effective in the rotation angle displacements. We show that the system is wellposed by using the Hille-Yosida theorem and prove that the system is exponentially stable if and only if the wave speeds are equal. Furthermore, we show that the system is polynomially stable provided that the wave speeds are not equal.


Introduction
With the increasing demand of advanced performance, the vibration suppression of the laminated beams has been one of the main research topics in smart materials and structures.These composite laminates usually have superior structural properties such as adaptability.The design of their piezoelectric materials can be used as both actuators and sensors [1].Hansen and Spies in [2] derived the mathematical model for two-layered beams with structural damping due to the interfacial slip, the system is given by the following equations: ρφ tt + G(ψ − φ x ) x = 0, (x, t) ∈ (0, 1) × (0, +∞), I ρ (3w − ψ) tt − G(ψ − φ x ) − D(3w − ψ) xx = 0, (x, t) ∈ (0, 1) × (0, +∞), 3I ρ w tt + 3G(ψ − φ x ) + 4γw + 4βw t − 3Dw xx = 0, (x, t) ∈ (0, 1) × (0, +∞), (1.1) where ρ, G, I ρ , D, γ, β are positive constant coefficients, ρ is the density of the beams, G is the shear stiffness, I ρ is the mass moment of inertia, D is the flexural rigidity, γ is the adhesive stiffness of the beams, and β is the adhesive damping parameter.The function φ denotes the transverse displacement of the beam which departs from its equilibrium position, ψ represents the rotation angle, w is proportional to the amount of slip along the interface at time t and longitudinal spatial variable x, 3w − ψ denotes the effective rotation angle, (1.1) 3 describes the dynamics of the slip.
In recent years, an increasing interest has been developed to determine the asymptotic behavior of the solution of several laminated beam problems.For example, Wang et al. [1] considered system (1.1) with the cantilever boundary conditions and two different wave speeds ( √ G/ρ and √ D/I ρ ).The authors proved the well-posedness and pointed out that system (1.1) can obtain the asymptotic stability but it does not reach the exponential stability due to the action of the slip w.Furthermore, to achieve the exponential decay result, the authors added an additional boundary control such that the boundary conditions become φ(0, t) = ξ(0, t) = w(0, t) = 0, w x (1, t) = 0, 3w(1, t) − ξ(1, t) − φ x (1, t) = u 1 (t) := k 1 φ t (1, t), ξ x (1, t) = u 2 (t) := −k 2 ξ t (1, t), where ξ = 3w − ψ.Cao et al. [3] considered the system (1.1) with following boundary conditions ψ(0, t) − φ x (0, t) = u 1 (t) := −k 1 φ t (0, t) − φ(0, t), (1, t), where ξ = 3w − ψ.The authors obtained an exponential stability result 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.Raposo [4] considered system (1.1) with two frictional dampings of the form (x, t) ∈ (0, 1) × (0, +∞), and obtained the exponential decay result under appropriate initial and boundary conditions.It is easy to find that if the slip w is assumed to be identically zero, then the first two equations of system (1.1) can be reduced exactly to the Timoshenko beam system.For the case of the Timoshenko beam with Fourier's law, many authors have shown various decay estimates depending on the wave speeds.Rivera and Racke [5] studied the Timoshenko system with thermoelastic dissipation, i.e., with positive constants ρ 1 , ρ 2 , ρ 3 , k, b, γ, κ.The authors showed that the exponential stability holds if and only if the wave speeds are equal ) . Júnior and Rivera [6] considered a new coupling to the thermoelastic Timoshenko beam of the form (1.4) The authors showed this system is exponentially stable if and only if the wave speeds are equal ) .On the contrary, the authors obtained the polynomially stable depending on the different boundary conditions.For system (1.4) with Dirichlet boundary conditions the authors obtained that the semigroup decay as 1 the authors obtained that the semigroup decay as 1 √ t .We refer the reader to [7,8,9,10,11,12,13,14,15,16,17], for some other related results.
Motivated by the above results, we intend to study the well-posedness and the asymptotic stability of the laminated beam system where the heat flux is given by Fourier's law.The system is written as where ρ, G, I ρ , D, σ, γ, β, k, τ are positive constant coefficients.We consider following initial and boundary conditions (1.6) By using Hille-Yosida theorem, we first prove the well-posedness result.By using the perturbed energy method, we then establish the exponential result if and only if ρ G = Iρ D and the polynomial stability if ρ G ̸ = Iρ D .Furthermore, by using Gearhart-Herbst-Prüss-Huang theorem, we obtain the lack of exponential stability.The main difficulty in carry out this paper is the appearance for the Fourier's law of heat conduction.For this purpose, we use the appropriated multiplies and energy method to build an equivalent Lyapunov functional.
We now briefly sketch the outline of the paper.In Section 2, we state and prove the wellposedness of problem (1.5)- (1.6).In Section 3, we establish an exponential stability result of the energy.In Section 4, the lack of exponential stability has been studied.Finally, Section 5 is devoted to the statement and proof of the polynomial stability.

The well-posedness
In this Section, we prove the well-posedness of problem (1.5)-(1.6)by using Hille-Yosida theorem.Firstly, we introduce the vector function Then system (1.5)-(1.6)can be written as where A is a linear operator defined by We consider the following spaces: equipped with the inner product Then, the domain of A is given by Proof.To obtain the above result, we need to prove that A : D(A ) → H is a maximal monotone operator.For this purpose, we need the following two steps: A is dissipative and Id − A is surjective.
Step 1.A is dissipative.For any U ∈ D(A ), by using the inner product and integration by parts, we can imply that Hence, A is a dissipative operator.
Step 2. Id − A is surjective.
To prove that the operator ( (2.5) 1 , (2.5) 3 and (2.5) (2.6) (2.7) Multiplying (2.7) 1 -(2.7) 4 by ṽ1 , ṽ3 , 3ṽ 5 and ṽ7 respectively, and integrating over (0, 1), we arrive at The sum of the equations in (2.8) gives the following variational formulation: where It is clear that a(•, •) and ã(•) are bounded.Furthermore, we can obtain that there exists a positive constant m such that which implies that a(•, •) is coercive.Hence, we assert that a(•, •) is a bilinear continuous coercive form on V × V , and ã(•) is a linear continuous form on V .Applying the Lax-Milgram theorem [18], we obtain that (2.8) has a unique solution (v 1 , v 3 , v 5 , v 7 ) T ∈ V.Then, by substituting v 1 , v 3 , v 5 into (2.6), we obtain for all ṽ1 ∈ H 1 * (0, 1), which implies Thus, by the L 2 theory for the linear elliptic equations, we obtain that Moreover, (2.10) is also true for any ϕ ∈ C 1 ([0, 1]) ⊂ H 1 * (0, 1) (ϕ(0) = 0).Hence, we get By using the integration by parts, we have In the same way, we get Finally, the application of the classical regularity theory for linear elliptic equations guarantees the existence of unique solution V ∈ D(A ) which satisfies (2.4).Hence, the operator Id − A is surjective.Moreover, it is easy to see that D(A ) is dense in H .

Exponential stability
In this Section, we prove the exponential decay for problem (1.5)-(1.6).It will be achieved by using the perturbed energy method.We define the energy functional E(t) as If the wave speeds are equal, we have the following exponentially stable result.
To prove our this result, we will state and prove some useful lemmas in advance.
Proof.By differentiating I 1 (t) with respect to t, using (1.5) 1 and integrating by parts, we obtain Then, we deduce that Making use of Young's inequality with ε 1 > 0, we obtain Then the estimate (3.9) is established.
Proof.Taking the derivative of I 5 (t) with respect to t, using (1.5) 2 and integrating by parts, we get Then, using Young's inequality, we arrive at (3.10).
Proof.By differentiating I 1 (t) with respect to t, using (1.5) 3 and integrating by parts, we obtain We then use Young's inequality with ε 3 > 0 to obtain (3.11).
Proof.By (1.5) 1 , (1.5) 3 and integrating by parts, we get We conclude for Gathering the estimates in the previous lemmas, we obtain At this point, we need to choose our constants very carefully.First, we choose ε 1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 small enough so that Then, we select δ 4 small enough so that Next, we choose δ 2 small enough so that Furthermore, we select δ 3 and δ 5 small enough so that Finally, we select δ 3 even smaller (if needed) and δ 1 small enough so that From the above, we deduce that there exist positive constants C 1 and C 2 such that (3.16) becomes By (3.3), we get for some positive constant C 3 .It is obvious that Recalling (3.18), we obtain for some positive constant c 1 .Then, a simple integration of (3.19) over (0, t) yields At last, estimate (3.20) gives the desired result (3.2) when combined with the equivalence of L(t) and E(t).

The lack of exponential stability
This Section is concerning the lack of exponential stability.Our result is achieved by Gearhart-Herbst-Prüss-Huang theorem to dissipative systems, see Prüss [21] and Huang [22].
Theorem 4.1 Let S(t) = e At be a C 0 -semigroup of contractions on Hilbert space H. Then S(t) is exponentially stable if and only if hold, where ρ(A) is the resolvent set of the differential operator A.
Next, we state and prove the main result of this section.Proof.We will prove that there exists a sequence of imaginary number λ µ and function with T not bounded.Rewrite spectral equation (4.1) in term of its components, we have for where λ ∈ R and F = (g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 ) T ∈ H. Taking g 1 = g 3 = g 5 = 0, then the above system becomes Because of the boundary conditions in (1.6), we can suppose that ) .

Now, choosing
Now, we take λ = λ µ such that ) 2 = 0, then the above system can be written as ) 2 ) Adding (4.5) 2 to (4.5) 3 , we get From (4.5) 4 , we get Substituting E into (4.6),we get where Substituting C into (4.5) 3 , we get Similarly, substituting C into (4.5) 1 , we get Substituting this expression into A, C and E, we obtain for µ → ∞, ) . Thus Therefore, there is no exponential stability.This completes the proof.

Polynomial stability
In this section, we consider the situation when the wave propagations are not the same.Proof.In this regard, we establish a polynomial decay result.As we will see, due to the presence of the ∫ 1 0 w tt φ x dx, we cannot directly perform the same proof as for the case where ρ G ̸ = Iρ D .To overcome this difficulty, the second-order energy method is needed.The second-order energy is defined by ) .
A simple calculation (Similar to (3.3)) implies that As in (3.14), we also define a Lyapunov functional L 2 (t) as follows: where I i (t), i = 1, 2, 3, 4 remain as defined in Lemma 3.3-Lemma 3.4 with derivatives of I 1 (t)-I 4 (t) remain the same while the derivative of I 5 (t) is given as for any ε 6 , ε 7 > 0. Observing Then, combining (5.4)-(5.5),we get Next, differentiating L 2 (t), we obtain At this point, we need to choose our constants very carefully.First, we choose ε 1 , ε 2 , ε 3 , ε 4 , ε 5 small enough so that Next, we choose δ 2 small enough so that Furthermore, we select δ 3 and δ 5 small enough so that Finally, we select δ 3 even smaller (if needed) and δ 1 small enough so that Thus, we deduce that there exist positive constants C (5.12) From the above, we deduce that there exist positive constant c such that (5.12) becomes L ′ 2 (t) ≤ − cE(t). (5.13) A simple integration of (5.13) over (0, t), recalling that E is non-increasing, yields Finally, for a positive constant c 2 , we have which completes the proof.

Theorem 3 . 1
Assume that ρ G = Iρ D hold.Let U 0 ∈ H , then there exists positive constants c 0 , c 1 such that the energy E(t) associated with problem (1.5)-(1.6)satisfies