Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 02, pp. 1–26. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DECAY RATES FOR TWO CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS OF TYPE III WITH INTERFACIAL SLIP AISSA GUESMIA Abstract. In this article we study the decay of solutions for two systems of laminated Timoshenko beams with interfacial slip, in the whole space R subject to a thermal effect of type III acting only on one component. When the thermal effect acts via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we prove that both systems are polynomially stable. Also we obtain stability estimates in the L2(R)-norm of solutions and their higher order derivatives with respect of the space variable. The decay rates, and the absence or presence of the regularity-loss type property, depend on the regularity of the initial data and the speeds of wave propagations. However, when the thermal effect acts via the first component (transversal displacement), we introduce a new stability number χ and prove that the stability of the system is equivalent to χ 6= 0. An application to a case of lower order coupling terms is also given. To prove our results, we use the energy method in the Fourier space combined with well chosen weight functions to build appropriate Lyapunov functionals. 1. Introduction A typical model of laminated Timoshenko beams of length L and with interfacial slip based on the Timoshenko theory can be formulated by the system (see [16, 17, 23] for more details) ρ1ϕtt + k(u− ϕx)x + F1 = 0, ρ2(3v − u)tt − b(3v − u)xx − k(u− ϕx) + F2 = 0, ρ̃3vtt − k̃0vxx + 3k(u− ϕx) + 4β̃vt + F̃3 = 0, (1.1) where the subscripts x and t denote the derivative with respect to space and time variables x and t, respectively, x ∈]0, L[ and t > 0, combining some initial data and boundary conditions at x = 0 and x = L. All the coefficients are positive constants and denote some physical properties of beams. The terms F1 = F1(x, t), F2 = F2(x, t) and F̃3 = F̃3(x, t) are external forces and play the role of controls. The functions ϕ = ϕ(x, t) and u = u(x, t) represent, respectively, the transverse and rotation angle displacements, and the function v = v(x, t) is proportional to 2010 Mathematics Subject Classification. 34B05, 34D05, 34H05. Key words and phrases. Timoshenko beam; interfacial slip; heat conduction; energy method; Fourier analysis. ©2022. This work is licensed under a CC BY 4.0 license. Submitted January 9, 2021. Published January 5, 2022. 1 2 A. GUESMIA EJDE-2022/02 the amount of slip along the interface, so the third equation in (1.1) describes the dynamics of the slip. Using the change of variables ρ3 = 1 9 ρ̃3, k1 = k, k2 = b, k3 = 1 9 k̃0, β = 4 9 β̃, w = −3v, ψ = 3v − u, F3 = 1 9 F̃3, the system (1.1) can be rewritten as ρ1ϕtt − k1(ϕx + ψ + w)x + F1 = 0, ρ2ψtt − k2ψxx + k1(ϕx + ψ + w) + F2 = 0, ρ3wtt − k3wxx + k1(ϕx + ψ + w) + βwt + F3 = 0. (1.2) This system is mathematically a particular case of the following more general one of Bresse-type ρ1ϕtt − k1(ϕx + ψ + lw)x − l̃k3(wx − l̃ϕ) + F1 = 0, ρ2ψtt − k2ψxx + k1(ϕx + ψ + lw) + F2 = 0, ρ3wtt − k3(wx − l̃ϕ)x + lk1(ϕx + ψ + lw) + βwt + F3 = 0, (1.3) where l and l̃ are positive constants. System (1.3) coincides with (1.2) when l = 1 and l̃ = 0. When w = F3 = l = l̃ = 0, system (1.3) is reduced to the Timoshenko- type system ρ1ϕtt − k1(ϕx + ψ)x + F1 = 0, ρ2ψtt − k2ψxx + k1(ϕx + ψ) + F2 = 0. (1.4) Systems (1.2), (1.3), and (1.4) were the subject of various studies in the literature during the previous thirty years, tackling well-posedness and stability questions by considering different types of controls Fj (dampings, memories, heat conduction effects, etc.). Let us mention here some of these studies related to our objectives in this paper. For the well-posedness and stability questions in the case of bounded domains, we refer the readers to the non exhaustive list of references [1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 21, 22, 23, 24, 25, 26, 27, 28, 34, 36]. We notice here that (1.2) was generally considered in the literature under the following restrictions: (1.2) is already damped via the control βwt and the speeds of the wave propagations of the last two equations in (1.2) are the same; that is, β > 0 and k2 ρ2 = k3 ρ3 . (1.5) For unbounded domains, the stability of (1.3) and (1.4) has been also treated in the literature for the previous few years. In this direction, we mention the papers [8, 11, 19, 20, 29, 31] (see also the references therein), where some polynomial stability estimates for L2(R)-norm of solutions were proved using frictional damping, heat conduction effects or memory controls. In this paper, we investigated the decay properties of two laminated Timoshenko beam with interfacial slip in the whole space R and without the restrictions (1.5). In addition, only one external force Fj is considered and it is generated by a thermal EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 3 effect of type III. Without loss of generality, the coefficients ρj in (1.2) are taken equal to 1. The first system we consider is the following ϕtt − k1(ϕx + ψ + w)x + τ1γqxt = 0, ψtt − k2ψxx + k1(ϕx + ψ + w) + τ2γqxt = 0, wtt − k3wxx + k1(ϕx + ψ + w) + τ3γqxt = 0, qtt − k4qxx − k5qxxt + γ(τ1ϕxt + τ2ψxt + τ3wxt) = 0, (1.6) where x ∈ R, t > 0, kj > 0, γ ∈ R∗, q = q(x, t) denotes the temperature and (τ1, τ2, τ3) ∈ {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. (1.7) The thermal dissipation in (1.6) is generated by the term −k5qxxt (see (2.9) in Section 2). In the second system of interest, the thermal dissipation is generated by the term of lower order k5qt; more precisely, we consider the system ϕtt − k1(ϕx + ψ + w)x + τ1γqxt = 0, ψtt − k2ψxx + k1(ϕx + ψ + w) + τ2γqxt = 0, wtt − k3wxx + k1(ϕx + ψ + w) + τ3γqxt = 0, qtt − k4qxx + k5qt + γ(τ1ϕxt + τ2ψxt + τ3wxt) = 0. (1.8) Systems (1.6) and (1.8) are subject to the initial conditions (ϕ,ψ,w, q)(x, 0) = (ϕ0, ψ0, w0, q0)(x), (ϕt, ψt, wt, qt)(x, 0) = (ϕ1, ψ1, w1, q1)(x). (1.9) The main objective of this article is to study the stability of (1.6) and (1.8) and to obtain some polynomial estimates in the L2(R)-norm of solutions and their higher order derivatives with respect to x. We will show that, when (τ1, τ2, τ3) = (1, 0, 0), both (1.6) and (1.8) are stable if and only if χ 6= 0, where χ := k3 − k2. (1.10) However, when (τ1, τ2, τ3) ∈ {(0, 1, 0), (0, 0, 1)}, (1.11) we prove that systems (1.6) and (1.8) are always stable, where the decay rate in the case k1 = k2 = k3 (1.12) is better than in the opposite one. Moreover, in the case (1.6), (1.12) allows to avoid the regularity restriction on the initial data known as the regularity-loss property (see [9, 18, 19, 30, 32, 33]). At the end of this article, we give an application to the case where the coupling terms between the laminated Timoshenko system and the equation of heat conduction in (1.6) and (1.8) τjγqxt and γ(τ1ϕxt + τ2ψxt + τ3wxt) (1.13) are, respectively, replaced by the following ones of lower order: τjγqt and − γ(τ1ϕt + τ2ψt + τ3wt). (1.14) Our stability results show that the effect of the heat conduction is better prop- agated to the whole system from the second or third equation of the laminated Timoshenko system than from the first one. The proof is based on the energy method combined with the Fourier analysis (by using the transformation in the Fourier space) and well chosen weight functions. 4 A. GUESMIA EJDE-2022/02 This article is organized as follows: in Section 2, we formulate (1.6) and (1.8) as a first order Cauchy system and give some preliminaries. In Section 3 we prove some differential identities. In Section 4, we prove our stability results. We end our paper by an application to the case (1.14) in Section 5. 2. Formulation of the problems We start by formulating (1.6) and (1.8) in an abstract first order system. To do so, we introduce the new variables u = ϕt, y = ψt, θ = wt, η = qt, v = ϕx + ψ + w, z = ψx, φ = wx and σ = qx. (2.1) Then systems (1.6) and (1.8) can be presented in the form vt − ux − y − θ = 0, ut − k1vx + τ1γηx = 0, zt − yx = 0, yt − k2 zx + k1v + τ2γ ηx = 0, φt − θx = 0, θt − k3 φx + k1v + τ3γηx = 0, σt − ηx = 0, ηt − k4σx + (1− k0)k5∂ k0 x η + γ(τ1ux + τ2yx + τ3θx) = 0, (2.2) where k0 = 2 in case (1.6), and k0 = 0 in case (1.8). Let U and its initial data U0 be given by U = (v, u, z, y, φ, θ, σ, η)T and U0 = (v, u, z, y, φ, θ, σ, η)T (·, 0). System (2.2) and the initial conditions (1.9) are reduced to Ut(x, t) +A2Uxx(x, t) +A1Ux(x, t) +A0U(x, t) = 0, U(x, 0) = U0(x), (2.3) where A2Uxx =  0 0 0 0 0 0 0 −ε0k5ηxx  , A1Ux =  −ux −k1vx + τ1γηx −yx −k2 zx + τ2γηx −θx −k3φx + τ3γηx −ηx −k4σx + γ(τ1ux + τ2yx + τ3θx)  , A0U =  −y − θ 0 0 k1v 0 k1v 0 (1− ε0)k5η  (2.4) EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 5 and ε0 = { 1 in case (1.6), 0 in case (1.8). (2.5) For a function h : R→ C, Reh, Imh, h̄ and ĥ denote the real part , the imaginary part, the conjugate, and the Fourier transformation of h, respectively. Using the Fourier transformation (with respect to the space variable x), (2.3) can be written in the Fourier space as the following first order Cauchy system Ût(ξ, t)− ξ2A2Û(ξ, t) + iξA1Û(ξ, t) +A0Û(ξ, t) = 0, ξ ∈ R, t > 0, Û(ξ, 0) = Û0(ξ), ξ ∈ R. (2.6) The solution of (2.6) is Û(ξ, t) = e−(−ξ 2A2+iξA1+A0)t Û0(ξ). (2.7) The energy Ê associated with (2.6) is Ê(ξ, t) = 1 2 [ k1|v̂|2 + |û|2 + k2|ẑ|2 + |ŷ|2 + k3|φ̂|2 + |θ̂|2 + k4|σ̂|2 + |η̂|2 ] . (2.8) System (2.6) is dissipative because d dt Ê(ξ, t) = −k5ξ2ε0 |η̂|2. (2.9) Indeed, the first equation in (2.6) is equivalent to v̂t − iξû− ŷ − θ̂ = 0, ût − ik1ξv̂ + iτ1γξη̂ = 0, ẑt − iξŷ = 0, ŷt − ik2ξẑ + k1v̂ + iτ2γξη̂ = 0, φ̂t − iξθ̂ = 0, θ̂t − ik3ξφ̂+ k1v̂ + iτ3γξη̂ = 0, σ̂t − iξη̂ = 0, η̂t − ik4ξσ̂ + k5ξ 2ε0 η̂ + iγξ(τ1û+ τ2ŷ + τ3θ̂). (2.10) To obtain (2.9), we multiply the equations in (2.10) by k1¯̂v, ¯̂u, k2¯̂z, ¯̂y, k3 ¯̂ φ, ¯̂ θ, k4 ¯̂σ, and ¯̂η, respectively. Then adding the obtained equations, taking the real part of the resulting expression and using the following classical relation, for two differentiable functions h, d : R→ C: d dt Re(hd̄) = Re(htd̄+ dth̄). (2.11) We observe that the energy Ê is equivalent to |Û |2 defined by |Û(ξ, t)|2 = |v̂|2 + |û|2 + |ẑ|2 + |ŷ|2 + |φ̂|2 + |θ̂|2 + |σ̂|2 + |η̂|2 because, for α1 = 1 2 min{k1, k2, k3, k4, 1} and α2 = 1 2 max{k1, k2, k3, k4, 1}, we have α1|Û(ξ, t)|2 ≤ Ê(ξ, t) ≤ α2|Û(ξ, t)|2, ∀ξ ∈ R, ∀t ∈ R+ . (2.12) Before presenting and proving our stability results in the next three sections, we prove these two lemmas that will be used in the proofs. 6 A. GUESMIA EJDE-2022/02 Lemma 2.1. Let r1, r2 and r3 be real numbers such that r1 > −1 and r2, r3 > 0. Then there exists Cr1,r2,r3 > 0 such that∫ 1 0 ξr1e−r3tξ r2 dξ ≤ Cr1,r2,r3(1 + t)−(r1+1)/r2 , ∀t ∈ R+. (2.13) Proof. For 0 ≤ t ≤ 1, (2.13) is evident, for any Cr1,r2,r3 ≥ 2(r1+1)/r2 r1+1 . For t > 1, we have ∫ 1 0 ξr1e−r3tξ r2 dξ = ∫ 1 0 ξr1+1−r2e−r3tξ r2 ξr2−1 dξ = ∫ 1 0 (ξr2)(r1+1−r2)/r2e−r3tξ r2 ξr2−1 dξ. Taking τ = r3tξ r2 , we have ξr2 = τ r3t and ξr2−1 dξ = 1 r2r3t dτ. Substituting in the above integral, we find∫ 1 0 (ξr2)(r1+1−r2)/r2e−r3tξ r2 ξr2−1 dξ = ∫ r3t 0 ( τ r3t )(r1+1−r2)/r2e−τ 1 r2r3t dτ ≤ 1 r2(r3t)(r1+1)/r3 ∫ +∞ 0 τ (r1+1−r2)/r2e−τ dτ ≤ 2(r1+1)/r2 r2r (r1+1)/r2 3 Cr1,r2(t+ 1)−(r1+1)/r2 , where Cr1,r2 = ∫ +∞ 0 τ (r1+1−r2)/r2e−τ dτ, which is a convergent integral, for any r1 > −1 and r2 > 0. This completes the proof of (2.13) with Cr1,r2,r3 = max {2(r1+1)/r2 r1 + 1 , 2(r1+1)/r2 r2r (r1+1)/r2 3 Cr1,r2 } . � Lemma 2.2. For any positive real numbers σ1, σ2, and σ3, we have sup |ξ|≥1 |ξ|−σ1e−σ2t|ξ|−σ3 ≤ (1 + σ1/(σ2σ3))σ1/σ3(1 + t)−σ1/σ3 , ∀t ∈ R+. (2.14) Proof. Clearly (2.14) is satisfied for t = 0. Let t > 0 and h(x) = x−σ1e−σ2t x −σ3 , for x ≥ 1. Simple computations show that h′(x) = (σ2σ3tx −σ3 − σ1)x−σ1−1e−σ2t x −σ3 . If t ≥ σ1/(σ2σ3), then h(x) ≤ h(((σ2σ3t)/σ1)1/σ3) = ((σ2σ3)/σ1)−σ1/σ3e−σ1/σ3(1 + 1/t)σ1/σ3(1 + t)−σ1/σ3 ≤ ((σ2σ3)/σ1)−σ1/σ3(1 + (σ2σ3)/σ1)σ1/σ3(1 + t)−σ1/σ3 = (1 + σ1/(σ2σ3))σ1/σ3(1 + t)−σ1/σ3 , EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 7 which gives (2.14) by taking x = |ξ|. If 0 < t < σ1/(σ2σ3), then h(x) ≤ h(1) = e−σ2t(1 + t)σ1/σ3(1 + t)−σ1/σ3 ≤ (1 + σ1/(σ2σ3))σ1/σ3(1 + t)−σ1/σ3 , which implies (2.14), for x = |ξ|. � 3. Preliminary differential identities This section is dedicated to the proof of several identities, which will play a crucial role in the proofs. In the rest of the paper, C and C̃ denote generic positive constants, and Cε denotes a generic positive constant depending on some positive constant ε. These generic constants can be different from line to line. Multiplying (2.10)4 and (2.10)3 by iξẑ and −iξŷ, respectively, and then adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re ( iξŷẑ ) = ξ2(|ŷ|2 − k2|ẑ|2)− k1 Re(iξv̂ẑ) + τ2γξ 2 Re(η̂ẑ). (3.1) Multiplying (2.10)2 and (2.10)1 by iξv̂, and −iξû, respectively,and then adding the resulting equations, taking the real part and using (2.11), we find d dt Re(iξûv̂) = ξ2(|û|2 − k1|v̂|2)− Re(iξŷû)− Re(iξθ̂û) + τ1γξ 2 Re(η̂v̂). (3.2) After, multiplying (2.10)6 and (2.10)5 by iξφ̂ and −iξθ̂, respectively, adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(iξθ̂φ̂) = ξ2(|θ̂|2 − k3|φ̂|2)− k1 Re(iξv̂φ̂) + τ3γξ 2 Re(η̂φ̂). (3.3) Multiplying (2.10)6 and (2.10)1 by −ξ2v̂ and −ξ2θ̂, respectively, then adding the resulting equations, taking the real part and using (2.11), we have d dt Re(−ξ2θ̂v̂) = ξ2(k1|v̂|2 − |θ̂|2)− ξ2 Re(iξûθ̂)− k3ξ2 Re(iξφ̂v̂) − ξ2 Re(ŷθ̂) + τ3γξ 2 Re t(iξη̂v̂). (3.4) Also, multiplying (2.10)4 and (2.10)1 by −ξ2v̂ and −ξ2ŷ, respectively, then adding the resulting equations, taking the real part and using (2.11), we infer that d dt Re(−ξ2ŷv̂) = ξ2(k1|v̂|2 − |ŷ|2)− ξ2 Re(iξûŷ)− k2ξ2 Re(iξẑv̂) − ξ2 Re(θ̂ŷ) + τ2γξ 2 Re(iξη̂v̂). (3.5) Multiplying (2.10)8 and (2.10)7 by iξσ̂ and −iξη̂, respectively, adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(iξη̂σ̂) = ξ2(|η̂|2 − k4|σ̂|2)− k5ξ2ε0 Re(iξη̂σ̂) + γξ2 Re(σ̂(τ1û+ τ2ŷ + τ3θ̂)). (3.6) Similarly, multiplying (2.10)3 and (2.10)6 by iξθ̂ and −iξẑ, respectively, then adding the resulting equations, taking the real part and using (2.11), we have d dt Re(iξẑθ̂) = −ξ2 Re(ŷθ̂) + k3ξ 2 Re(φ̂ẑ) + k1 Re(iξv̂ẑ)− τ3γξ2 Re(η̂ẑ). (3.7) 8 A. GUESMIA EJDE-2022/02 Multiplying (2.10)5 and (2.10)4 by iξŷ and −iξφ̂, respectively, then adding the resulting equations, taking the real part and using (2.11), we arrive at d dt Re(iξφ̂ŷ) = −ξ2 Re(θ̂ŷ) + k2ξ 2 Re(ẑφ̂) + k1 Re(iξv̂φ̂)− τ2γξ2 Re(η̂φ̂). (3.8) Multiplying (2.10)2 and (2.10)3 by −ẑ and −û, respectively, then adding the re- sulting equations, taking the real part and using (2.11), it follows that d dt Re(−ûẑ) = −k1 Re(iξv̂ẑ)− Re(iξŷû) + τ1γ Re(iξη̂ẑ). (3.9) Finally, multiplying (2.10)2 and (2.10)5 by −φ̂ and −û, respectively, then adding the resulting equations, taking the real part and using (2.11), it follows that d dt Re(−ûφ̂) = −k1 Re(iξv̂φ̂)− Re(iξθ̂û) + τ1γ Re(iξη̂φ̂). (3.10) 4. Stability In this section, we investigate the asymptotic behavior, when time t goes to infin- ity, of the solution U of (2.3). First, we will show that |Û |2 converges exponentially to zero (with respect to time t) in case (1.11), and in case (τ1, τ2, τ3) = (1, 0, 0) with χ 6= 0. In case (τ1, τ2, τ3) = (1, 0, 0) with χ = 0, we prove that |Û |2 does not converge to zero when t goes to infinity. Let us distinguish the three cases (1.7). Case 1.1: (τ1, τ2, τ3) = (1, 0, 0) and χ 6= 0. We start by presenting the exponential stability result for (2.6) in the next lemma. Lemma 4.1. Assume that χ 6= 0; that is k2 6= k3. Let Û be a solution of (2.6). Then there exist c, c̃ > 0 such that |Û(ξ, t)|2 ≤ c̃e−cf(ξ)t|Û0(ξ)|2, ∀ξ ∈ R, ∀t ∈ R+, (4.1) where f(ξ) = ξ4+2ε0 f̃(ξ) and f̃(ξ) = { 1 + ξ8 in case (1.6), 1 + ξ6 in case (1.8). (4.2) Proof. Multiplying (2.10)2 and (2.10)8 by i |γ|γ ξη̂ and −i |γ|γ ξû, respectively, the n adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re ( i |γ| γ ξûη̂ ) = |γ|ξ2(|η̂|2 − |û|2) + |γ| γ k4ξ 2 Re(σ̂û) − |γ| γ k1ξ 2 Re(v̂η̂) + |γ| γ k5ξ 2ε0 Re(iξη̂û). (4.3) Multiplying (2.10)6 and (2.10)8 by η̂ and θ̂, respectively, adding the resulting equa- tions, taking the real part and using (2.11), we find that d dt Re(η̂θ̂) = γ Re(iξθ̂û) + k4 Re(iξσ̂θ̂)− k5ξ2ε0 Re(η̂θ̂) + k3 Re(iξφ̂η̂)− k1 Re(v̂η̂). (4.4) EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 9 Also, multiplying (2.10)4 and (2.10)8 by η̂ and ŷ, respectively, adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(η̂ŷ) = −γ Re(iξûŷ) + k4 Re(iξσ̂ŷ)− k5ξ2ε0 Re(η̂ŷ) + k2 Re(iξẑη̂)− k1 Re(v̂η̂). (4.5) Multiplying (2.10)1 and (2.10)7 by iξσ̂ and −iξv̂, respectively, then adding the resulting equations, taking the real part and using (2.11), we infer that d dt Re(iξv̂σ̂) = −ξ2 Re(σ̂û) + ξ2 Re(v̂η̂) + Re(iξŷσ̂) + Re(iξθ̂σ̂). (4.6) Similarly, multiplying (2.10)3 and (2.10)7 by −σ̂ and −ẑ, respectively, then adding the resulting equations, taking the real part and using (2.11), we arrive at d dt Re(−σ̂ẑ) = Re(iξσ̂ŷ) + Re(iξẑη̂). (4.7) Multiplying (2.10)5 and (2.10)7 by −σ̂ and −φ̂, respectively, then adding the re- sulting equations, taking the real part and using (2.11), we entail d dt Re(−σ̂φ̂) = Re(iξσ̂θ̂) + Re(iξφ̂η̂). (4.8) Let λ0, . . . , λ5 be positive constants to be defined later, and let (observe that χ 6= 0 by assumption) λ6 = k2 χ (λ4 + λ5), λ7 = −k3 χ (λ4 + λ5), λ8 = k2 k1 λ5ξ 2 − λ1 + k2 χ (λ4 + λ5), λ9 = k3 k1 λ4ξ 2 − λ3 − k3 χ (λ4 + λ5). We define the functional F0(ξ, t) = Re [ iξ (λ1ŷẑ + λ2ûv̂ + λ3θ̂φ̂+ η̂σ̂ + λ6ẑθ̂ + λ7φ̂ŷ) ] + Re ( − λ4ξ2θ̂v̂ − λ5ξ2ŷv̂ − λ8ûẑ − λ9ûφ̂ ) . (4.9) Multiplying (3.1)-(3.10) by λ1, . . . , λ5, 1, λ6, . . . , λ9, respectively, and then adding the obtained equations, we see that, thanks to the choices of λ6, . . . , λ9, the expres- sion of d dtF0 does not contain the terms Re(iξv̂ẑ), Re(iξv̂φ̂), Re(ŷθ̂), and Re(φ̂ẑ) because their coefficients vanish. So, we find that d dt F0(ξ, t) = −ξ2 ( k3λ3|φ̂|2 + (λ5 − λ1)|ŷ|2 + (λ4 − λ3)|θ̂|2 + (k1λ2 − k1λ4 − k1λ5)|v̂|2 ) − ξ2(k2λ1|ẑ|2 + k4|σ̂|2) + I1 Re(iξθ̂û) + I2 Re(iξŷû) + γξ2 Re(σ̂û) + ξ2(λ2|û|2 + |η̂|2) + Re ( iγλ8ξη̂ẑ + iγλ9ξη̂φ̂− ik5ξ2ε0+1η̂σ̂ + γλ2ξ 2η̂v̂ ) , (4.10) where I1 = λ4ξ 2 − λ2 − λ9 and I2 = λ5ξ 2 − λ2 − λ8. (4.11) 10 A. GUESMIA EJDE-2022/02 To eliminate the terms Re(iξθ̂û), Re(iξŷû), and Re(σ̂û) from the right hand side of (4.10), we put I3 = γ + |γ| γ k4λ0 + k4 γ I1, I4 = γ + |γ| γ k4λ0 + k4 γ I2, I5 = γ + |γ| γ k4λ0, and introduce the functional F1(ξ, t) = F0(ξ, t) + |γ| γ λ0 Re(iξûη̂)− 1 γ Re(I1η̂θ̂ + I2η̂ŷ) + I5 Re(iξv̂σ̂)− I3 Re(σ̂φ̂)− I4 Re(σ̂ẑ). (4.12) Multiplying (4.3)-(4.8) by λ0, − 1 γ I1, − 1 γ I2, I5, I4 and I3, respectively, and then adding the obtained equations and (4.10), we arrive at d dt F1(ξ, t) = −ξ2 ( k2λ1|ẑ|2 + k3λ3|φ̂|2 + (λ5 − λ1)|ŷ|2 + (λ4 − λ3)|θ̂|2 + (k1λ2 − k1λ4 − k1λ5)|v̂|2 ) − ξ2((|γ|λ0 − λ2)|û|2 + k4|σ̂|2) + (|γ|λ0 + 1)ξ2|η̂|2 + Re(i |γ| γ k5λ0ξ 2ε0+1η̂û− ik5ξ2ε0+1η̂σ̂) + Re [(k1 γ (I1 + I2) + ( γλ2 + γ + |γ| γ (k4 − k1)λ0 ) ξ2 ) η̂v̂ + k5 γ I1ξ 2ε0 η̂θ̂ + k5 γ I2ξ 2ε0 η̂ŷ ] + Re [ i ( γλ8 + k2 γ I2 − I4 ) ξη̂ẑ + i ( γλ9 + k3 γ I1 − I3 ) ξη̂φ̂ ] . (4.13) Let λ be a positive constant. We introduce the functionals (f̃ is defined in (4.2)) F (ξ, t) = ξ2+2ε0F1(ξ, t) and L(ξ, t) = λÊ(ξ, t) + 1 f̃(ξ) F (ξ, t). (4.14) For the rest of proofs, we will frequently use the inequality |ξ|m2 ≤ |ξ|m1 + |ξ|m3 , ∀ξ ∈ R, ∀0 ≤ m1 ≤ m2 ≤ m3. (4.15) According to (4.15), we observe that |Ij | ≤ C(ξ2 + 1), j = 1, 2, 3, 4. Then, applying Young’s inequality for the terms depending on η̂ in (4.13), it follows, for any ε > 0, that d dt F (ξ, t) ≤ −ξ4+2ε0 ( (k2λ1 − ε)|ẑ|2 + (k3λ3 − ε)|φ̂|2 + (λ5 − λ1 − ε)|ŷ|2 + (λ4 − λ3 − ε)|θ̂|2 ) − ξ4+2ε0 ( (k1λ2 − k1λ4 − k1λ5 − ε)|v̂|2 + (|γ|λ0 − λ2 − ε)|û|2 + (k4 − ε)|σ̂|2 ) + Cε,λ0,...,λ9 f̃(ξ)ξ2ε0 |η̂|2. (4.16) We choose λ1, λ3 > 0, then we select λ0 such that λ0 > 1 |γ| (λ1 +λ3). After, we pick λ4 and λ2 such that λ3 < λ4 < |γ|λ0 − λ1 and λ1 + λ4 < λ2 < |γ|λ0. Finally, we take λ5 and ε such that λ1 < λ5 < λ2 − λ4 and 0 < ε < min { λ5 − λ1, k1(λ2 − λ4 − λ5), λ4 − λ3, |γ|λ0 − λ2, k2λ1, k3λ3, k4 } . EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 11 Hence, using the definition (2.8) of Ê, (4.16) leads to, for some positive constant c1, d dt F (ξ, t) ≤ −c1ξ4+2ε0Ê(ξ, t) + Cf̃(ξ)ξ2ε0 |η̂|2. (4.17) Thus, from (2.9), (4.14), and (4.17), we have d dt L(ξ, t) ≤ −c1f(ξ)Ê(ξ, t)− (k5λ− C)ξ2ε0 |η̂|2, (4.18) where f is defined in (4.2). Moreover, using the definitions of Ê, F , L, and f̃ , we obtain, for some c2 > 0 (independent of λ), |L(ξ, t)− λÊ(ξ, t)| = 1 f̃(ξ) |F (ξ, t)| ≤ C (1 + ξ2)ξ2+2ε0 f̃(ξ) ≤ c2Ê(ξ, t). (4.19) Therefore, for λ large enough so that λ > max{ Ck5 , c2}, we deduce from (4.18) and (4.19) that d dt L(ξ, t) + c1f(ξ)Ê(ξ, t) ≤ 0, (4.20) c3Ê(ξ, t) ≤ L(ξ, t) ≤ c4Ê(ξ, t), (4.21) where c3 = λ− c2 > 0 and c4 = λ+ c2 > 0. Consequently, a combination of (4.20) and the second inequality in (4.21) lead to, for c = c1 c4 , d dt L(ξ, t) + cf(ξ)L(ξ, t) ≤ 0. (4.22) Finally, by integration (4.22) with respect to time t and using (2.12) and (4.21), (4.1) follows with c̃ = c4α2 c3α1 . � Theorem 4.2. Assume that χ 6= 0; that is k2 6= k3. Let N, ` ∈ N such that ` ≤ N , U0 ∈ HN (R) ∩ L1(R) and U be the solution of (2.3). Then for any j ∈ {0, . . . , N − `}, there exists c0 > 0 such that ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/12−j/6‖U0‖L1(R) + c0(1 + t)−`/2‖∂j+`x U0‖L2(R), (4.23) for all t ∈ R+ in case (1.6), and ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/8−j/4‖U0‖L1(R) + c0(1 + t)−`/2‖∂j+`x U0‖L2(R), (4.24) for all t ∈ R+ in case (1.8). Proof. From (4.2) we have in case (1.6) (low and high frequencies) f(ξ) ≥ { ξ6/5 if |ξ| ≤ 1, ξ−2/5 if |ξ| > 1. (4.25) 12 A. GUESMIA EJDE-2022/02 Applying Plancherel’s theorem and (4.1), we have ‖∂jxU‖2L2(R) = ‖∂̂jxU(x, t)‖2L2(R) = ∫ R ξ2 j |Û(ξ, t)|2dξ ≤ c̃ ∫ R ξ2 je−cf(ξ)t|Û0(ξ)|2dξ ≤ c̃ ∫ |ξ|≤1 ξ2 je−cf(ξ)t|Û0(ξ)|2dξ + c̃ ∫ |ξ|>1 ξ2je−cf(ξ)t|Û0(ξ)|2 dξ := J1 + J2. (4.26) Using (2.13) (with r1 = 2j, r3 = c 5 and r2 = 6) and (4.25), it follows, for the low frequency region, J1 ≤ C‖Û0‖2L∞(R) ∫ |ξ|≤1 ξ2 je− c 5 tξ 6 dξ ≤ C(1 + t)− 1 6 (1+2 j)‖U0‖2L1(R). (4.27) For the high frequency region, using (4.25), we observe that J2 ≤ C ∫ |ξ|>1 |ξ|2 je− c5 tξ −2 |Û(ξ, 0)|2 dξ ≤ C sup |ξ|>1 {|ξ|−2 ` e− c5 t|ξ| −2 } ∫ R |ξ|2(j+`)|Û(ξ, 0)|2 dξ, then, using (2.14) (with σ1 = 2l, σ2 = c 5 and σ3 = 2), J2 ≤ C(1 + t)−`‖ ∂j+`x U0‖2L2(R), (4.28) and so, by combining (4.26)–(4.28), we obtain (4.23). The proof of (4.24) is very similar; we notice only, in case (1.8), that f(ξ) ≥ { ξ4/4 if |ξ| ≤ 1, ξ−2/4 if |ξ| > 1. � Remark 4.3. It is well known that the behavior of the Fourier transform of U in the low frequency region determines the rate of decay of U , while its behavior in the high frequency region imposes a regularity restriction on the initial data known as the regularity- loss property; see [9, 18, 19, 30, 32, 33]. The fact that f tends to 0 when ξ goes to infinity leads to the regularity-loss property in the estimates on ‖∂jxU‖L2(R) because (4.23) and (4.24) with j = ` = 0 imply only the boundedness of ‖U‖L2(R). This remark is valid also in case (1.11) for (1.8), and in case (1.11) for (1.6) if (1.12) is not satisfied (see Theorem 4.6 and Theorem 4.9 below). Case 1.2: (τ1, τ2, τ3) = (1, 0, 0) and χ = 0. In this subsection, we prove that (2.6) is not stable if (τ1, τ2, τ3) = (1, 0, 0) and χ = 0. Theorem 4.4. Assume that χ = 0; that is k2 = k3. Then |Û(ξ, t)| does not converge to zero when time t goes to infinity. Proof. We show that, for any ξ ∈ R, the matrix A := −(−ξ2A2 + iξA1 +A0) (4.29) has at least a pure imaginary eigenvalue; that is ∀ξ ∈ R, ∃λ ∈ C : Re(λ) = 0, Im(λ) 6= 0 and det(λI −A) = 0, (4.30) EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 13 where I denotes the identity matrix. From (2.4) with (τ1, τ2, τ3) = (1, 0, 0) and k2 = k3, we have λI −A =  λ −iξ 0 −1 0 −1 0 0 −ik1ξ λ 0 0 0 0 0 iγξ 0 0 λ −iξ 0 0 0 0 k1 0 −ik2ξ λ 0 0 0 0 0 0 0 0 λ −iξ 0 0 k1 0 0 0 −ik2ξ λ 0 0 0 0 0 0 0 0 λ −iξ 0 iγξ 0 0 0 0 −ik4ξ k5ξ 2ε0 + λ  . A direct computation shows that det(λI −A) = 2k1λ 2(λ2 + k2ξ 2) [ λ(λ+ k5ξ 2ε0) + (k4 + γ2)ξ2 ] + k4ξ 2(λ2 + k1ξ 2)(λ2 + k2ξ 2)2 + λ(λ2 + k2ξ 2)2 [ λ2(λ+ k5ξ 2ε0) + γ2λξ2 + k1ξ 2(λ+ k5ξ 2ε0) ] . It is clear that, if ξ 6= 0, then λ = i √ k2ξ is a pure imaginary eigenvalue of A. If ξ = 0, then λ = i √ 2k1 is a pure imaginary eigenvalue of A. Consequently, according to (2.7) and (4.29) (see [35]), the solution of (2.6) does not converge to zero when time t goes to infinity. � Case 2: (τ1, τ2, τ3) = (0, 1, 0). We present, first, our exponential stability result for (2.6), where the proof is similar to the one of Lemma 4.1. Lemma 4.5. Let Û be a solution of (2.6). Then there exist c, c̃ > 0 such that (4.1) is satisfied with f(ξ) = ξ4+2ε0 f̃(ξ) , f̃(ξ) =  1 + ξ6 for (1.6) and (1.8) under (1.12), 1 + ξ10 for (1.6) without (1.12), 1 + ξ8 for (1.8) without (1.12). (4.31) Proof. Multiplying (2.10)4 and (2.10)8 by i |γ|γ ξη̂ and −i |γ|γ ξŷ, respectively, then adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(i |γ| γ ξŷη̂) = |γ|ξ2(|η̂|2 − |ŷ|2) + |γ| γ k4ξ 2 Re(σ̂ŷ)− |γ| γ k1 Re(iξv̂η̂) − |γ| γ k2ξ 2 Re(η̂ẑ) + |γ| γ k5ξ 2ε0 Re(iξη̂ŷ). (4.32) Multiplying (2.10)1 and (2.10)7 by ξ2σ̂ and ξ2v̂, respectively, then adding the re- sulting equations, taking the real part and using (2.11), we find that d dt Re(ξ2v̂σ̂) = ξ2 Re(σ̂ŷ) + ξ2 Re(σ̂θ̂) + ξ2 Re(iξûσ̂) + ξ2 Re(iξη̂v̂). (4.33) Also, multiplying (2.10)3 and (2.10)7 by iξσ̂ and −iξẑ, respectively, adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(iξẑσ̂) = −ξ2 Re(σ̂ŷ) + ξ2 Re(η̂ẑ). (4.34) 14 A. GUESMIA EJDE-2022/02 Multiplying (2.10)2 and (2.10)8 by η̂ and û, respectively, then adding the resulting equations, taking the real part and using (2.11), we infer that d dt Re(ûη̂) = −γ Re(iξŷû) + k1 Re(iξv̂η̂) + k4 Re(iξσ̂û)− k5ξ2ε0 Re(η̂û). (4.35) Multiplying (2.10)5 and (2.10)7 by iξσ̂ and −iξφ̂, respectively, then adding the resulting equations, taking the real part and using (2.11), we see that d dt Re(iξφ̂σ̂) = −ξ2 Re(σ̂θ̂) + ξ2 Re(η̂φ̂). (4.36) Finally, multiplying (2.10)6 and (2.10)8 by −iξη̂ and iξθ̂, respectively, adding the resulting equations, taking the real part and using (2.11), it follows that d dt Re(iξη̂θ̂) = γξ2 Re(ŷθ̂)− k4ξ2 Re(σ̂θ̂) + k3ξ 2 Re(η̂φ̂)− k5ξ2ε0 Re(iξη̂θ̂) + k1 Re(iξv̂η̂). (4.37) Let λ0, . . . , λ5 be positive constants, and let λ6 = k2 k3 [(k3 k1 − 1 ) λ4ξ 2 − λ2 − λ3 ] , λ7 = −k3 k2 λ6, λ8 = −k2 k1 λ5ξ 2 + λ6 − λ1, λ9 = λ4ξ 2 + λ2. We define the functional F0(ξ, t) = Re [ iξ ( λ1ŷẑ − λ2ûv̂ + λ3θ̂φ̂+ η̂σ̂ + λ6ẑθ̂ + λ7φ̂ŷ )] + Re ( − λ4ξ2θ̂v̂ + λ5ξ 2ŷv̂ − λ8ûẑ − λ9ûφ̂ ) . (4.38) Multiplying (3.1)-(3.10) by λ1, −λ2, λ3, λ4, −λ5, 1, λ6, . . . , λ9, respectively, and adding the resulting equations, we find that d dt F0(ξ, t) = −ξ2 ( k3λ3|φ̂|2 + λ2|û|2 + (λ4 − λ3)|θ̂|2 + (k1λ5 − k1λ4 − k1λ2)|v̂|2 ) − ξ2(k2λ1|ẑ|2 + k4|σ̂|2) + I1 Re(iξŷû) + I2ξ 2 Re(ŷθ̂) + γξ2 Re(σ̂ŷ) + ξ2((λ1 + λ5)|ŷ|2 + |η̂|2) + Re ( γλ1ξ 2η̂ẑ − γλ7ξ2η̂φ̂− ik5ξ2ε0+1η̂σ̂ − iγλ5ξ3η̂v̂ ) (4.39) (thanks to the choices of λ6, . . . , λ9, Re(iξ(v̂ẑ + θ̂û + v̂φ̂)) and Re(ẑφ̂) disappear), where I1 = −λ5ξ2 + λ2 − λ8 and I2 = λ5 − λ4 − λ6 − λ7. We put I3 = ( |γ| γ k4λ0 + γ ) ξ2 + k4 γ I1 and I4 = k4 γ (I2ξ 2 + I1), EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 15 and introduce the functional F1(ξ, t) = ξ2F0(ξ, t) + |γ| γ λ0ξ 2 Re(iξŷη̂) + k4 γ I1ξ 2 Re(v̂σ̂) + I3 Re(iξẑσ̂) + 1 γ I1ξ 2 Re(ûη̂) + I4 Re(iξφ̂σ̂)− 1 γ I2ξ 2 Re(iξη̂θ̂). (4.40) Multiplying (4.32)-(4.37) and (4.39) by λ0ξ 2, k4 γ I1, I3, 1 γ I1ξ 2, I4, − 1 γ I2ξ 2 and ξ2, respectively, then adding the obtained expressions, we arrive at (Re(iξûσ̂) and Re(σ̂ŷ + σ̂θ̂) disappear according to the definition of I3 and I4) d dt F1(ξ, t) = −ξ4(k2λ1|ẑ|2 + k3λ3|φ̂|2 + λ2|û|2 + (λ4 − λ3)|θ̂|2 + (k1λ5 − k1λ4 − k1λ2)|v̂|2)− ξ4((|γ|λ0 − λ1 − λ5)|ŷ|2 + k4|σ̂|2) + (|γ|λ0 + 1)ξ4|η̂|2 + ξ2 Re [ (iI5v̂ + I6ẑ + I7φ̂− ik5ξ2ε0+1σ̂ + i |γ| γ k5λ0ξ 2ε0+1ŷ + i k5 γ ξ2ε0+1I2θ̂ − k5 γ ξ2ε0I1û)η̂ ] , (4.41) where I5 = −γλ5ξ3 + ( |γ| γ k1λ0 + k4 − k1 γ I1 + k1 γ I2)ξ, I6 = (−|γ| γ k2λ0 + γλ1)ξ2 + I3, I7 = −( k3 γ I2 + γλ7)ξ2 + I4. Observe that, by definition, |I1| ≤ { C if (1.12) holds, C(1 + ξ2) if not, |I2| ≤ { C if (1.12) holds, C(1 + ξ2) if not, (4.42) |I5| ≤ C(|ξ|+ |ξ|3), |I6| ≤ C(1 + ξ2), |I7| ≤ { C(1 + ξ2) if (1.12) holds, C(1 + ξ4) if not. (4.43) Then, applying Young’s inequality, it follows, for any ε > 0, that ξ2 Re [( iI5v̂ + I6ẑ + I7φ̂− ik5ξ2ε0+1σ̂ + i |γ| γ k5λ0ξ 2ε0+1ŷ + i k5 γ ξ2ε0+1I2θ̂ − k5 γ ξ2ε0I1û ) η̂ ] ≤ εξ4 ( |ẑ|2 + |φ̂|2 + |û|2 + |θ̂|2 + |v̂|2 + |σ̂|2 + |ŷ|2 ) + Cε(ξ 4ε0 |I1|2 + ξ4ε0+2|I2|2 + |I5|2 + |I6|2 + |I7|2 + ξ4ε0+2)|η̂|2 ≤ εξ4 ( |ẑ|2 + |φ̂|2 + |û|2 + |θ̂|2 + |v̂|2 + |σ̂|2 + |ŷ|2 ) + Cε,λ0,...,λ9 f̃(ξ)|η̂|2. (4.44) 16 A. GUESMIA EJDE-2022/02 By combining (4.41) and (4.44), we find that d dt F1(ξ, t) ≤ −ξ4 ( (k2λ1 − ε)|ẑ|2 + (k3λ3 − ε)|φ̂|2 + (λ2 − ε)|û|2 + (λ4 − λ3 − ε)|θ̂|2 ) − ξ4 ( (k1λ5 − k1λ4 − k1λ2 − ε)|v̂|2 + (|γ|λ0 − λ1 − λ5 − ε)|ŷ|2 + (k4 − ε)|σ̂|2 ) + Cε,λ0,...,λ9 f̃(ξ)|η̂|2. (4.45) Let λ be a positive constant. We introduce the functionals F (ξ, t) = ξ2ε0F1(ξ, t) and L(ξ, t) = λÊ(ξ, t) + 1 f̃(ξ) F (ξ, t). (4.46) We choose 0 < λ1, 0 < λ3 < λ4 < λ5, 0 < λ2 < λ5 − λ4, λ0 > 1 |γ| (λ1 + λ5) and 0 < ε < min { k2λ1, k3λ3, λ2, λ4 − λ3, k1λ5 − k1λ4 − k1λ2, |γ|λ0 − λ1 − λ5, k4 } , and use the definition of Ê, we deduce from (4.45) and (4.46), for some positive constant c1, that d dt F (ξ, t) ≤ −c1ξ4+2ε0Ê(ξ, t) + Cf̃(ξ)ξ2ε0 |η̂|2. (4.47) Then, from (2.9), (4.46) and (4.47), we infer that d dt L(ξ, t) ≤ −c1f(ξ)Ê(ξ, t)− (k5λ− C)ξ2ε0 |η̂|2. (4.48) On the other hand, the definitions of Ê, F and L imply that there exists c2 > 0 (independent of λ) such that, for d0 = 0 if (1.12) holds, and d0 = 5 if not,∣∣L(ξ, t)− λÊ(ξ, t) ∣∣ ≤ c2 ξ2ε0(1 + ξ4 + |ξ|d0) f̃(ξ) Ê(ξ, t) ≤ 6c2Ê(ξ, t). So, we choose λ > max{ Ck5 , 6c2}, we obtain (4.20) and (4.21) with c3 = λ− 6c2 > 0 and c4 = λ+ 6c2 > 0. The proof can be ended as for Lemma 4.1. � Theorem 4.6. Let N, ` ∈ N such that ` ≤ N , U0 ∈ HN (R) ∩ L1(R) and U be the solution of (2.3). Then for any j ∈ {0, . . . , N − `}, there exist c0, c̃0 > 0 such that, for any t ∈ R+, (i) Case (1.6): ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/12−j/6‖U0‖L1(R) + c0e −c̃0t‖∂jxU0‖L2(R) (4.49) if k1 = k2 = k3, and ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/12−j/6‖U0‖L1(R) + c0(1 + t)−`/4‖∂j+`x U0‖L2(R) (4.50) if not. (ii) Case (1.8): ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/8−j/4‖U0‖L1(R) + c0(1 + t)−`/2‖∂j+`x U0‖L2(R) (4.51) if k1 = k2 = k3, and ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/8−j/4‖U0‖L1(R) + c0(1 + t)−`/4‖∂j+`x U0‖L2(R) (4.52) if not. EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 17 Proof. For (1.6), from (4.31) (low and high frequencies) we have: if k1 = k2 = k3, then f(ξ) ≥ { ξ6/4 if |ξ| ≤ 1, 1/4 if |ξ| > 1 ; (4.53) otherwise f(ξ) ≥ { ξ6/6 if |ξ| ≤ 1, ξ−4/6 if |ξ| > 1 . (4.54) The proof of (4.50) is identical to the one of Theorem 4.2 by using (4.54) and applying (2.13) (with r1 = 2j, r2 = c 6 and r3 = 6) and (2.14) (with σ1 = 2l, σ2 = c 6 and σ3 = 4). To obtain (4.49), noticing that the low frequencies can be treated as for (4.50). For the high frequencies, we observe that (4.53) implies that∫ |ξ|>1 |ξ|2 je−cf(ξ)t|Û(ξ, 0)|2 dξ ≤ ∫ |ξ|>1 |ξ|2 je−ct/4|Û(ξ, 0)|2 dξ ≤ e−ct/4 ∫ R |ξ|2 j |Û(ξ, 0)|2 dξ ≤ e−ct/4‖ ∂jxU0‖2L2(R), so (4.49) holds with c̃0 = c 8 . The proof of (4.51) and (4.52) is identical to the one of (4.50) by remarking, for (1.8), that: if k1 = k2 = k3, then f(ξ) ≥ { ξ4/4 if |ξ| ≤ 1, ξ−2/4 if |ξ| > 1 ; otherwise f(ξ) ≥ { ξ4/5 if |ξ| ≤ 1, ξ−4/5 if |ξ| > 1 . � Remark 4.7. In case (1.6) under (1.12), the fact that f tends to 1 when ξ goes to infinity allows to avoid the regularity-loss property in the estimate (4.49) on ‖∂jxU‖L2(R) because one can take j = ` = 0, and the stability of (2.3) is still satisfied with a decay estimate depending only on ‖U0‖L1(R) and ‖U0‖L2(R). This remark is valid also for (1.6) in case (τ1, τ2, τ3) = (0, 0, 1) under (1.12) (see Theorem 4.9 below). Case 3: (τ1, τ2, τ3) = (0, 0, 1). In this case, we prove the same stability results for (2.6) and (2.3) that given in the previous subsection, and moreover, the proofs are very similar. Lemma 4.8. The result of Lemma 4.5 holds when (τ1, τ2, τ3) = (0, 0, 1). Proof. Multiplying (2.10)6 and (2.10)8 by i |γ|γ ξη̂ and −i |γ|γ ξθ̂, respectively, then adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(i |γ| γ ξθ̂η̂) = |γ|ξ2(|η̂|2 − |θ̂|2) + |γ| γ k4ξ 2 Re(σ̂θ̂)− |γ| γ k1 Re(iξv̂η̂) − |γ| γ k3ξ 2 Re(η̂φ̂) + |γ| γ k5ξ 2ε0 Re(iξη̂θ̂). (4.55) 18 A. GUESMIA EJDE-2022/02 Also, multiplying (2.10)2 and (2.10)8 by η̂ and û, respectively, adding the resulting equations, taking the real part and using (2.11), we infer that d dt Re(ûη̂) = −γ Re(iξθ̂û) + k1 Re(iξv̂η̂) + k4 Re(iξσ̂û)− k5ξ2ε0 Re(η̂û). (4.56) Finally, multiplying (2.10)4 and (2.10)8 by −iξη̂ and iξŷ, respectively, then adding the resulting equations, taking the real part and using (2.11), it follows that d dt Re(iξη̂ŷ) = γξ2 Re(ŷθ̂)− k4ξ2 Re(σ̂ŷ) + k2ξ 2 Re(η̂ẑ) − k5ξ2ε0 Re(iξη̂ŷ) + k1 Re(iξv̂η̂). (4.57) Let λ0, . . . , λ5 be positive constants, and let λ6 = k2 k3 [( 1− k3 k1 ) λ4ξ 2 − λ2 − λ3 ] , λ7 = −k3 k2 λ6, λ8 = k2 k1 λ5ξ 2 + λ6 − λ1, λ9 = −λ4ξ2 + λ2. We define the functional F0(ξ, t) = Re [ iξ (λ1ŷẑ − λ2ûv̂ + λ3θ̂φ̂+ η̂σ̂ + λ6ẑθ̂ + λ7φ̂ŷ) ] + Re(λ4ξ 2θ̂v̂ − λ5ξ2ŷv̂ − λ8ûẑ − λ9ûφ̂). (4.58) Multiplying (3.1)-(3.10) by λ1,−λ2, λ3,−λ4, λ5, 1, λ6, . . . , λ9, respectively, and then adding the resulting equations, we find that d dt F0(ξ, t) = −ξ2(k3λ3|φ̂|2 + λ2|û|2 + (λ5 − λ1)|ŷ|2 + (k1λ4 − k1λ5 − k1λ2)|v̂|2) − ξ2(k2λ1|ẑ|2 + k4|σ̂|2) + I1 Re(iξθ̂û) + I2ξ 2 Re(ŷθ̂) + γξ2 Re(σ̂θ̂) + ξ2((λ3 + λ4)|θ̂|2 + |η̂|2) + Re ( γλ3ξ 2η̂φ̂− γλ6ξ2η̂ẑ − ik5ξ2ε0+1η̂σ̂ − iγλ4ξ3η̂v̂ ) , (4.59) where I1 = λ5ξ 2 + λ2 − λ8 and I2 = λ4 − λ5 − λ6 − λ7. We put I3 = ( |γ| γ k4λ0 + γ ) ξ2 + k4 γ I1 and I4 = k4 γ (I2ξ 2 + I1), and introduce the functional F1(ξ, t) = ξ2F0(ξ, t) + |γ| γ λ0ξ 2 Re(iξθ̂η̂) + k4 γ I1ξ 2 Re(v̂σ̂) + I3 Re(iξẑσ̂) + 1 γ I1ξ 2 Re(ûη̂) + I4 Re(iξφ̂σ̂)− 1 γ I2ξ 2 Re(iξη̂ŷ). (4.60) Multiplying (4.55), (4.33), (4.34), (4.56), (4.36), (4.57), and (4.59) by λ0ξ 2, k4γ I1, I4, 1 γ I1ξ 2, I3, − 1 γ I2ξ 2 and ξ2, respectively, and then adding the obtained expressions, EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 19 we arrive at (observe that (4.33), (4.34) and (4.36) are valid also in case (τ1, τ2, τ3) = (0, 0, 1)) d dt F1(ξ, t) = −ξ4 ( k2λ1|ẑ|2 + k3λ3|φ̂|2 + λ2|û|2 + (λ5 − λ1)|ŷ|2 + (k1λ5 − k1λ4 − k1λ2)|v̂|2 ) − ξ4 ( (|γ|λ0 − λ3 − λ4)|θ̂|2 + k4|σ̂|2 ) + (|γ|λ0 + 1)ξ4|η̂|2 + ξ2 Re [ (iI5v̂ + I6φ̂+ I7ẑ − ik5ξ2ε0+1σ̂ + i |γ| γ k5λ0ξ 2ε0+1θ̂ + i k5 γ ξ2ε0+1I2ŷ − k5 γ ξ2ε0I1û)η̂ ] , (4.61) where I5 = −γλ4ξ3 + ( |γ| γ k1λ0 + k4 − k1 γ I1 + k1 γ I2)ξ, I6 = (−|γ| γ k3λ0 + γλ3)ξ2 + I3, I7 = −( k2 γ I2 + γλ6)ξ2 + I4. We see that (4.42) and (4.43) are still valid. Then, applying Young’s inequality, we obtain (4.44). Therefore, we define F and L by (4.46) and choose 0 < λ3, 0 < λ1 < λ4 < λ5, 0 < λ2 < λ5 − λ4, λ0 > 1 |γ| (λ3 + λ4) and 0 < ε < min{k2λ1, k3λ3, λ2, λ5 − λ1, k1λ5 − k1λ4 − k1λ2, |γ|λ0 − λ3 − λ4, k4}, we obtain (4.47) and (4.48). Consequently, the proof can be ended as for Lemma 4.5. � Theorem 4.9. The stability result in Theorem 4.6 is satisfied when (τ1, τ2, τ3) = (0, 0, 1). The proof of the above theorem is identical to the one of Theorem 4.6; therefore we omit it. 5. Application: lower order coupling terms (1.14) This section concerns the stability of (2.3) in case where the coupling terms (1.13) are replaced by the ones (1.14); more precisely, we study the stability of ϕtt − k1(ϕx + ψ + w)x + τ1γqt = 0, ψtt − k2ψxx + k1(ϕx + ψ + w) + τ2γqt = 0, wtt − k3wxx + k1(ϕx + ψ + w) + τ3γqt = 0, qtt − k4qxx − k5qxxt − γ(τ1ϕt + τ2ψt + τ3wt) = 0 (5.1) and ϕtt − k1(ϕx + ψ + w)x + τ1γqt = 0, ψtt − k2ψxx + k1(ϕx + ψ + w) + τ2γqt = 0, wtt − k3wxx + k1(ϕx + ψ + w) + τ3γqt = 0, qtt − k4qxx + k5qt − γ(τ1ϕt + τ2ψt + τ3wt) = 0 (5.2) 20 A. GUESMIA EJDE-2022/02 with the initial conditions (1.9). We define U , its initial data U0 and the energy Ê as in Section 2. It is clear that (2.3), (2.6), (2.7), and (2.9) are valid with A2 as in (2.4), A1Ux =  −ux −k1vx −yx −k2 zx −θx −k3 φx −ηx −k4σx  , A0U =  −y − θ τ1γη 0 k1v + τ2γη 0 k1v + τ3γη 0 (1− ε0)k5η − γ(τ1u+ τ2y + τ3θ)  . (5.3) So, instead of (2.10), we have v̂t − iξû− ŷ − θ̂ = 0, ût − ik1ξv̂ + τ1γη̂ = 0, ẑt − iξŷ = 0, ŷt − ik2ξẑ + k1v̂ + τ2γη̂ = 0, φ̂t − iξθ̂ = 0, θ̂t − ik3ξφ̂+ k1v̂ + τ3γη̂ = 0, σ̂t − iξη̂ = 0, η̂t − ik4ξσ̂ + k5ξ 2ε0 η̂ − γ (τ1û+ τ2ŷ + τ3θ̂). (5.4) Lemma 5.1. Let Û be a solution of (2.6). Then (i) If (τ1, τ2, τ3) = (1, 0, 0) and χ = 0, |Û(ξ, t)| doesn’t converge to zero when time t goes to infinity. (ii) There exist c, c̃ > 0 such that (4.1) holds true with the following f : Case (τ1, τ2, τ3) = (1, 0, 0) and χ 6= 0: f(ξ) = ξ4+2ε0 f̃(ξ) and f̃(ξ) = { 1 + ξ10 for (5.1), 1 + ξ8 for (5.2). (5.5) Case (τ1, τ2, τ3) ∈ {(0, 1, 0), (0, 0, 1)}: f(ξ) = ξ2+2ε0 f̃(ξ) and f̃(ξ) =  1 + ξ6 for (5.1) under (1.12), 1 + ξ4 for (5.2) under (1.12), 1 + ξ10 for (5.1) without (1.12), 1 + ξ8 for (5.2) without (1.12). (5.6) Proof. The proof is very similar to the one given in Sections 3 and 4 with some small modifications related to the coupling terms (1.14). We give here a brief idea of the proof. We see that, for (5.4), the expressions (3.1)-(3.5) and (3.7)-(3.10) are satisfied with τjγη̂ instead of iτjγξη̂, and (3.6) holds true if we replace iγξ(τ1û+ τ2ŷ+ τ3θ̂) by −γ(τ1û+ τ2ŷ + τ3θ̂). Now, we distinguish the cases (τ1, τ2, τ3) = (1, 0, 0), (τ1, τ2, τ3) = (0, 1, 0) and (τ1, τ2, τ3) = (0, 0, 1). EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 21 Case 1.1: (τ1, τ2, τ3) = (1, 0, 0) and χ 6= 0. We start by modifying the expres- sions (4.3)-(4.8) (according to (5.4)). Multiplying (5.4)2 and (5.4)8 by − |γ|γ ξ 2η̂ and − |γ|γ ξ 2û, respectively, then adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(−|γ| γ ξ2ûη̂) = |γ|ξ2(|η̂|2 − |û|2)− |γ| γ k4ξ 2 Re(iξσ̂û) − |γ| γ k1ξ 2 Re(iξv̂η̂) + |γ| γ k5ξ 2ε0+2 Re(η̂û). (5.7) Multiplying (5.4)6 and (5.4)8 by −iξη̂ and iξθ̂, respectively, adding the resulting equations, taking the real part and using (2.11), we find d dt Re ( iξη̂θ̂ ) = γ Re(iξûθ̂)− k4ξ2 Re(σ̂θ̂)− k5ξ2ε0 Re(iξη̂θ̂) + k3ξ 2 Re(φ̂η̂) + k1 Re(iξv̂η̂). (5.8) Also, multiplying (5.4)4 and (5.4)8 by −iξη̂ and iξŷ, respectively, adding the re- sulting equations, taking the real part and using (2.11), we obtain d dt Re(iξη̂ŷ) = γ Re(iξûŷ)− k4ξ2 Re(σ̂ŷ)− k5ξ2ε0 Re(iξη̂ŷ) + k2ξ 2 Re(ẑη̂) + k1 Re(iξv̂η̂). (5.9) Multiplying (5.4)1 and (5.4)7 by σ̂ and v̂, respectively, adding the resulting equa- tions, taking the real part and using (2.11), we infer that d dt Re(v̂σ̂) = −Re(iξσ̂û) + Re(iξv̂η̂) + Re(ŷσ̂) + Re(θ̂σ̂). (5.10) Similarly, multiplying (5.4)3 and (5.4)7 by iξσ̂ and −iξẑ, respectively, then adding the resulting equations, taking the real part and using (2.11), we arrive at d dt Re(iξẑσ̂) = −ξ2 Re(σ̂ŷ) + ξ2 Re(ẑη̂). (5.11) Multiplying (5.4)5 and (5.4)7 by iξσ̂ and −iξφ̂, respectively, adding the resulting equations, taking the real part and using (2.11), we entail d dt Re(iξφ̂σ̂) = −ξ2 Re(σ̂θ̂) + ξ2 Re(φ̂η̂). (5.12) We put F̃0(ξ, t) = ξ2F0(ξ, t), where F0 is defined in (4.9). Multiplying (3.1)-(3.10) (with the modifications cited above) by λ1, . . . , λ5, 1, λ6, . . . , λ9, respectively, and adding the obtained expressions, we find (instead of (4.10)) d dt F̃0(ξ, t) = −ξ4(k3λ3|φ̂|2 + (λ5 − λ1)|ŷ|2 + (λ4 − λ3)|θ̂|2 + (k1λ2 − k1λ4 − k1λ5)|v̂|2)− ξ4(k2λ1|ẑ|2 + k4|σ̂|2) + I1ξ 2 Re(iξθ̂û) + I2ξ 2 Re(iξŷû)− γξ2 Re(iξσ̂û) + ξ4(λ2|û|2 + |η̂|2) + ξ2 Re ( γλ8η̂ẑ + γλ9η̂φ̂− ik5ξ2ε0+1η̂σ̂ − iγλ2ξη̂v̂ ) , (5.13) 22 A. GUESMIA EJDE-2022/02 where I1 and I2 are defined in (4.11). We put I3 = −k4 γ (I1 + |γ|λ0)ξ2− γ, I4 = −k4 γ (I2 + |γ|λ0)ξ2− γ, I5 = −γ− |γ| γ k4λ0ξ 2, and introduce the functional F1(ξ, t) = F̃0(ξ, t)− |γ| γ λ0ξ 4 Re(ûη̂) + 1 γ ξ2 Re(iI1ξη̂θ̂ + iI2ξη̂ŷ) + I5ξ 2 Re(v̂σ̂) + I4 Re(iξẑσ̂) + I3 Re(iξφ̂σ̂). (5.14) Multiplying (5.7)-(5.12) by λ0ξ 2, 1 γ I1ξ 2, 1 γ I2ξ 2, I5ξ 2, I4 and I3, respectively, and then adding the obtained equations and (5.13), we arrive at d dt F1(ξ, t) = −ξ4 ( k2λ1|ẑ|2 + k3λ3|φ̂|2 + (λ5 − λ1)|ŷ|2 + (λ4 − λ3)|θ̂|2 + (k1λ2 − k1λ4 − k1λ5)|v̂|2 ) − ξ4((|γ|λ0 − λ2)|û|2 + k4|σ̂|2) + (|γ|λ0 + 1)ξ4|η̂|2 + ξ2 Re ( |γ| γ k5λ0ξ 2ε0+2η̂û − ik5ξ2ε0+1η̂σ̂ ) + ξ2 Re [ i( k1 γ (I1 + I2) + γλ2 + I5 − |γ| γ k1λ0ξ 2)ξv̂η̂ + i k5 γ I1ξ 2ε0+1θ̂η̂ + i k5 γ I2ξ 2ε0+1ŷη̂ ] + ξ2 Re [( γλ8 + I4 + k2 γ I2ξ 2 ) η̂ẑ + ( γλ9 + I3 + k3 γ I1ξ 2 ) η̂φ̂ ] . (5.15) Now, we consider f̃ and f defined in (5.5), and introduce the functionals F (ξ, t) = ξ2ε0F1(ξ, t) and L(ξ, t) = λÊ(ξ, t) + 1 f̃(ξ) F (ξ, t). (5.16) Applying Young’s inequality, (5.15) implies (4.16). So, the proof can be completed as for Lemma 4.1. Case 1.2: (τ1, τ2, τ3) = (1, 0, 0) and χ = 0. To prove that |Û(ξ, t)| does not converge to zero when time t goes to infinity, it is enough to prove (4.30), where (according to (5.4)) λI −A =  λ −iξ 0 −1 0 −1 0 0 −ik1ξ λ 0 0 0 0 0 γ 0 0 λ −iξ 0 0 0 0 k1 0 −ik2ξ λ 0 0 0 0 0 0 0 0 λ −iξ 0 0 k1 0 0 0 −ik2ξ λ 0 0 0 0 0 0 0 0 λ −iξ 0 −γ 0 0 0 0 −ik4ξ k5ξ 2ε0 + λ  . A direct computation shows that det(λI −A) = 2k1λ 2(λ2 + k2ξ 2)[λ(λ+ k5ξ 2ε0) + k4ξ 2 + γ2] + k4ξ 2(λ2 + k1ξ 2)(λ2 + k2ξ 2)2 + λ(λ2 + k2ξ 2)2[λ2(λ+ k5ξ 2ε0) + γ2λ+ k1ξ 2(λ+ k5ξ 2ε0)]. Then, the conclusions indicated in the proof of Theorem 4.4 are valid for (5.4). EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 23 Case 2: (τ1, τ2, τ3) = (0, 1, 0). First, we modify the expressions (4.32)-(4.37) ac- cording to (5.4). Multiplying (5.4)4 and (5.4)8 by− |γ|γ ξ 2η̂ and− |γ|γ ξ 2ŷ, respectively, then adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re ( − |γ| γ ξ2ŷη̂ ) = |γ|ξ2(|η̂|2 − |ŷ|2)− |γ| γ k4ξ 2 Re(iξσ̂ŷ) + |γ| γ k1ξ 2 Re(η̂v̂) − |γ| γ k2ξ 2 Re(iξẑη̂) + |γ| γ k5ξ 2ε0+2 Re(η̂ŷ). (5.17) Multiplying (5.4)1 and (5.4)7 by iξσ̂ and −iξv̂, respectively, adding the resulting equations, taking the real part and using (2.11), we find d dt Re(iξv̂σ̂) = −Re(iξσ̂ŷ)− Re(iξσ̂θ̂)− ξ2 Re(ûσ̂) + ξ2 Re(η̂v̂). (5.18) Also, multiplying (5.4)3 and (5.4)7 by σ̂ and ẑ, respectively, adding the resulting equations, taking the real part and using (2.11), we obtain d dt Re(ẑσ̂) = −Re(iξσ̂ŷ) + Re(iξη̂ẑ). (5.19) Multiplying (5.4)2 and (5.4)8 by −iξη̂ and iξû, respectively, adding the resulting equations, taking the real part and using (2.11), we infer that d dt Re(−iξûη̂) = γ Re(iξŷû) + k1ξ 2 Re(v̂η̂)− k4ξ2 Re(σ̂û)− k5ξ2ε0 Re(iξη̂û). (5.20) Multiplying (5.4)5 and (5.4)7 by σ̂ and φ̂, respectively, adding the resulting equa- tions, taking the real part and using (2.11), we see that d dt Re(φ̂σ̂) = −Re(iξσ̂θ̂) + Re(iξη̂φ̂). (5.21) Finally, multiplying (5.4)6 and (5.4)8 by η̂ and θ̂, respectively, adding the resulting equations, taking the real part and using (2.11), it follows that d dt Re(η̂θ̂) = γ Re(ŷθ̂) + k4 Re(iξσ̂θ̂)− k3 Re(iξη̂φ̂)− k5ξ2ε0 Re(η̂θ̂)− k1 Re(v̂η̂). (5.22) We define the functional F0 by (4.38), and we obtain (instead of (4.39)) d dt F0(ξ, t) = −ξ2(k3λ3|φ̂|2 + λ2|û|2 + (λ4 − λ3)|θ̂|2 + (k1λ5 − k1λ4 − k1λ2)|v̂|2) − ξ2(k2λ1|ẑ|2 + k4|σ̂|2) + I1 Re(iξŷû) + I2ξ 2 Re(ŷθ̂)− γ Re(iξσ̂ŷ) + ξ2((λ1 + λ5)|ŷ|2 + |η̂|2) + Re(−iγλ1ξη̂ẑ + iγλ7ξη̂φ̂ − ik5ξ2ε0+1η̂σ̂ − γλ5ξ2η̂v̂). (5.23) We put I3 = −|γ| γ k4λ0ξ 2 − k4 γ I1 − γ and I4 = −k4 γ (I2ξ 2 + I1), 24 A. GUESMIA EJDE-2022/02 and introduce the functional F1(ξ, t) = F0(ξ, t)− |γ| γ λ0ξ 2 Re(ŷη̂) + k4 γ I1 Re(iξv̂σ̂) + I3 Re(ẑσ̂) + 1 γ I1 Re(iξûη̂) + I4 Re(φ̂σ̂)− 1 γ I2ξ 2 Re(η̂θ̂). (5.24) Multiplying (5.17)-(5.22) and (5.23) by λ0, k4 γ I1, I3, − 1 γ I1, I4, − 1 γ I2ξ 2 and 1, respectively, and adding the obtained expressions, we arrive at d dt F1(ξ, t) = −ξ2 ( k2λ1|ẑ|2 + k3λ3|φ̂|2 + λ2|û|2 + (λ4 − λ3)|θ̂|2 + (k1λ5 − k1λ4 − k1λ2)|v̂|2 ) − ξ2((|γ|λ0 − λ1 − λ5)|ŷ|2 + k4|σ̂|2) + (|γ|λ0 + 1)ξ2|η̂|2 + ξRe [ (I5v̂ + iI6ẑ + iI7φ̂− ik5ξ2ε0 σ̂ + |γ| γ k5λ0ξ 2ε0+1ŷ + k5 γ ξ2ε0+1I2θ̂ + i k5 γ ξ2ε0I1û)η̂ ] , (5.25) where I5 = ( |γ| γ k1λ0 − γλ5 + k1 γ (I2 − I1) + k4 γ I1)ξ, I6 = |γ| γ k2λ0ξ 2 − γλ1 + I3, I7 = k3 γ I2ξ 2 + I4 + γλ7. Because (4.42) is still satisfied, we infer that, for f̃ defined in (5.6), d dt F1(ξ, t) ≤ −ξ2((k2λ1 − ε)|ẑ|2 + (k3λ3 − ε)|φ̂|2 + (λ2 − ε)|û|2 + (λ4 − λ3 − ε)|θ̂|2) − ξ2((k1λ5 − k1λ4 − k1λ2 − ε)|v̂|2 + ( |γ|λ0 − λ1 − λ5 − ε ) |ŷ|2 + (k4 − ε)|σ̂|2) + Cε,λ0,...,λ9 f̃(ξ)|η̂|2. (5.26) Therefore, we introduce the functionals F and L defined in (4.46) and consider the same choices of λ0, . . . , λ5 and ε, we arrive at d dt F (ξ, t) ≤ −c1ξ2+2ε0Ê(ξ, t) + Cf̃(ξ)ξ2ε0 |η̂|2. (5.27) Hence, the proof can be completed as for Lemma 4.1. Case 3: (τ1, τ2, τ3) = (0, 0, 1). This case can be treated using very similar modifi- cations to the ones considered for the case (τ1, τ2, τ3) = (0, 1, 0); we omit the details here. � Theorem 5.2. Let N, ` ∈ N such that ` ≤ N , U0 ∈ HN (R) ∩ L1(R) and U be the solution of (2.3). Then for any j ∈ {0, . . . , N − `}, there exist c0, c̃0 > 0 such that, for any t ∈ R+, (i) Case (τ1, τ2, τ3) = (1, 0, 0) and χ 6= 0: ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/12−j/6‖U0‖L1(R) + c0(1 + t)−`/4‖∂j+`x U0‖L2(R) EJDE-2022/02 CAUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEMS 25 for (5.1), and ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/8−j/4‖U0‖L1(R) + c0(1 + t)−`/4‖∂j+`x U0‖L2(R) for (5.2). (ii) Case (τ1, τ2, τ3) ∈ {(0, 1, 0), (0, 0, 1)} and k1 = k2 = k3: ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/8−j/4‖U0‖L1(R) + c0(1 + t)−`/2‖∂j+`x U0‖L2(R) for (5.1), and ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/4−j/2‖U0‖L1(R) + c0(1 + t)−`/2‖∂j+`x U0‖L2(R) for (5.2). (iii) Case (τ1, τ2, τ3) ∈ {(0, 1, 0), (0, 0, 1)} and k1 = k2 = k3: ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/8−j/4‖U0‖L1(R) + c0(1 + t)−`/6‖∂j+`x U0‖L2(R) for (5.1), and ‖∂jxU‖L2(R) ≤ c0(1 + t)−1/4−j/2‖U0‖L1(R) + c0(1 + t)−`/6‖∂j+`x U0‖L2(R) for (5.2). The proof of the above theorem is identical to the one of Theorem 4.6; therefore we omit it. Acknowledgments. The author would like to express his gratitude to the anony- mous referees for their objective suggestions, which allowed to improve this article. References [1] M. S. Alves, P. Gamboa, G. C. Gorain, A. Rambaud, O. Vera; Asymptotic behavior of a flexible structure with Cattaneo type of thermal effect, Indagationes Mathematicae, 27 (2016), 821-834. [2] T. A. Apalara; Uniform stability of a laminated beam with structural damping and second sound, ZAMP, 68 (2017), 40-55. [3] T. A. Apalara; On the stability of thermoelastic laminated beam, Acta. Math. Scie., 39 (2019), 1517-1524. [4] C. F. Beards, I. M. A. Imam; The damping of plate vibration by interfacial slip between layers, Int. J. Mach. Tool. Des. Res., 18 (1978), 131-137. [5] X. G. Cao, D. Y. Liu, G. Q. Xu; Easy test for stability of laminated beams with structural damping and boundary feedback controls, phJ. Dynamical Control Syst., 13 (2007), 313-336. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, F. A. Falcao Nascimento, I. Lasiecka, H. Rodrigues; Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, ZAMP, 65 (2014), 1189-1206. [7] Z. Chen, W. Liu, D. Chen; General decay rates for a laminated beam with memory, Taiw. J. Math., 23 (2019), 1227-1252. [8] L. Djouamai, B. Said-Houari; A new stability number of the Bresse-Cattaneo system, Math. Meth. Appl. Sci., 41 (2018), 2827-2847. [9] L. H. Fatori, R. N. Monteiro, H. D. Fernández Sare; The Timoshenko system with history and Cattaneo law, Applied Mathematics and Computation, 228 (2014), 128-140. [10] B. Feng, T. E. Ma, R. N. Monteiro, C. A. Raposo; Dynamics of laminated Timoshenko beams, J. Dyn. Diff. Equa., 30 (2018), 1489-1507. [11] T. E. Ghoul, M. Khenissi, B. Said-Houari; On the stability of the Bresse system with frictional damping, J. Math. Anal. Appl., 455 (2017), 1870-1898. [12] A. Guesmia; Asymptotic stability of Bresse system with one infinite memory in the longitu- dinal displacements, Medi. J. Math., 14 (2017), 19 pages. [13] A. Guesmia; Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement, Nonauton. Dyn. Syst., 4 (2017), 78-97. 26 A. GUESMIA EJDE-2022/02 [14] A. Guesmia; Well-posedness and stability results for laminated Timoshenko beams with in- terfacial slip and infinite memory, IMA J. Math. Cont. Info., 37 (2020), 300-350. [15] A. Guesmia, S. Messaoudi, A. Soufyane; On the stabilization for a linear Timoshenko system with infinite history and applications to the coupled Timoshenko-heat systems, Elec. J. Diff. Equa., 2012 (2012), 1-45. [16] S. W. Hansen; In control and estimation of distributed parameter systems: Non-linear phe- nomena, International Series of Numerical Analysis, 118 (1994), 143-170. [17] S. W. Hansen, R. Spies; Structural damping in a laminated beams due to interfacial slip, J. Sound Vibration, 204 (1997), 183-202. [18] K. Ide, K. Haramoto, S. Kawashima; Decay property of regularity-loss type for dissipative Timoshenko system, Math. Mod. Meth. Appl. Sci., 18 (2008), 647-667. [19] M. Khader, B. Said-Houari; Decay rate of solutions to Timoshenko system with past history in unbounded domains, Appl. Math. Optim., 75 (2017), 403-428. [20] M. Khader, B. Said-Houari; Optimal decay rate of solutions to Timoshenko system with past history in unbounded domains, Z. Anal. Anwend, 37 (2018), 435-459. [21] G. Li, X. Kong, W. Liu; General decay for a laminated beam with structural damping and memory: the case of non-equal wave speeds, J. Inte. Equa., 30 (2018), 95-116. [22] W. Liu, W. Zhao; Exponential and polynomial decay for a laminated beam with Fourier’s type heat conduction, Preprints 2017, 2017020058, doi: 10.20944/preprints201702.0058.v1. [23] A. Lo, N. E Tatar; Stabilization of laminated beams with interfacial slip, Elec. J. Diff. Equa., 2015 (2015), 1-14. [24] A. Lo, N. E. Tatar; Uniform stability of a laminated beam with structural memory, Qual. Theory Dyn. Syst., 15 (2016), 517-540. [25] A. Lo, N. E. Tatar; Exponential stabilization of a structure with interfacial slip, Discrete Contin. Dyn. Syst., 36 (2016), 6285-6306. [26] M. I. Mustafa; Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl., 466 (2018), 619-641. [27] C. A. Raposo; Exponential stability for a structure with interfacial slip and frictional damping, Appl. Math. Lett., 53 (2016), 85-91. [28] C. A. Raposo, O. V. Villagrán, J. E. Muñoz Rivera, M. S. Alves; Hybrid laminated Timo- shenko beam, J. Math. Phys., 58 (2017), 11 pages. [29] B. Said-Houari; R. Racke; Decay rates and global existence for semilinear dissipative Timo- shenko systems, Quart. Appl. Math., 71 (2013), 229-266. [30] B. Said-Houari, R. Rahali; Asymptotic behavior of the Cauchy problem of the Timoshenko system in thermoelsaticity of type III, Evol. Equa. Cont. Theory, 2 (2013), 423-440. [31] B. Said-Houari, A. Soufyane; The effect of frictional damping terms on the decay rate of the Bresse system, Evol. Equa. Cont. Theory, 3 (2014), 713-738. [32] M. L. Santos, D. S. Almeida, J. E. Muñoz Rivera; The stability number of the Timoshenko system with second sound, J. Diff. Equa., 253 (2012), 2715-2733. [33] A. Soufyane, B. Said-Houari; The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equa. Cont. Theory, 3 (2014), 713-738. [34] N. E. Tatar; Stabilization of a laminated beam with interfacial slip by boundary controls, Boundary Value Problem, 2015, DOI: 10.1186/s13661-015-0432-3. [35] G. Teschl; Ordinary differential equations and dynamical systems, Amer. Math. Soc., 140 (2012), ISBN 978-0-8218-8328-0. [36] J. M. Wang, G. Q. Xu, S. P. Yung; Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44 (2005), 1575-1597. Aissa Guesmia Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, 3 Rue Augustin Fresnel, BP 45112, 57073 Metz Cedex 03, France Email address: aissa.guesmia@univ-lorraine.fr 1. Introduction 2. Formulation of the problems 3. Preliminary differential identities 4. Stability Case 1.1: (1 ,2 ,3)=(1,0,0) and =0 Case 1.2: (1 ,2 ,3)=(1,0,0) and = 0 Case 2: (1 ,2 ,3)=(0,1,0) Case 3: (1 ,2 ,3)=(0,0,1) 5. Application: lower order coupling terms (??) Case 1.1: (1 ,2 ,3)=(1,0,0) and =0. Case 1.2: (1 ,2 ,3)=(1,0,0) and = 0 Case 2: (1 ,2 ,3)=(0,1,0) Case 3: (1 ,2 ,3)=(0,0,1) Acknowledgments References