Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 121, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu STABILIZATION OF COUPLED THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS SABEUR MANSOURI, LOUIS TEBOU Abstract. We consider a coupled system consisting of a Kirchhoff thermoe- lastic plate and an undamped wave equation. It is known that the Kirchhoff thermoelastic plate is exponentially stable. The coupling is weak. First, we show that the coupled system is not exponentially stable. Afterwards, we prove that the coupled system is polynomially stable, and provide an explicit polyno- mial decay rate of the associated semigroup. Our proof relies on a combination of the frequency domain method and the multipliers technique. 1. Introduction Since the pioneering work of Dafermos [14] on the stability of the thermoelas- ticity equations in the late sixties, followed by the book of Lagnese [23] on the boundary stabilization of thin elastic plates, there has been a tremendous amount of activity involving the stabilization of thermoelastic systems, especially since the nineties. It was known since the work of Dafermos that the semigroup generated by the infinitesimal operator of the thermoelasticity equations is not even strongly stable, except for certain geometric configurations. It then made sense for Lagnese to tackle the exponential stability problem for a thermoelastic plate by adding me- chanical damping mechanisms on a suitable portion of the boundary [23, Chap. 7]. Then arose the natural question of whether for thermoelastic plates, the presence of mechanical damping on the boundary was necessary or not. In other words, could one dispense of the extra mechanical damping, and still exponentially stabi- lize a thermoelastic plate by relying solely on the dissipation induced by the heat component of the system? A first answer to that challenging question was pro- vided by Kim, who proved that no mechanical damping was necessary to ensure the exponential stability of a clamped plate [21]. Later on, Liu and Renardy [32] proved that the semigroup associated with a clamped or hinged thermoelastic plate is analytic, which is a stronger notion than exponential stability for strongly stable semigroups. Then followed many other works in the same vein by, e.g. Liu and Liu [30], Lasiecka and Triggiani [24, 25, 26], Lasiecka and Avalos [6, 7, 8], Zuazua and collaborators [11, 37, 48], Munõz Rivera and collaborators [15, 33, 34]. As those stabilization works on thermoelastic plates were being carried out, Lebeau and Zuazua returned to the stability of the thermoelasticity equations, and they 2010 Mathematics Subject Classification. 93D20, 35L05, 47D06, 47N70, 74F05, 74K20. Key words and phrases. Kirchhoff thermoelastic plate; wave equation; stabilization; weakly coupled equations; frequency domain method; multipliers technique. c©2020 Texas State University. Submitted January 28, 2020. Published December 16, 2020. 1 2 S. MANSOURI, L. TEBOU EJDE-2020/121 proved exponential decay to a finite dimensional subspace and polynomial stability under certain geometric constraints [28]. Other closely related works include, e.g. [4, 13, 42]. In the present work, we are interested in answering the following question: Know- ing that the clamped Kirchhoff thermoelastic plate is exponentially stable, e.g. [7, 15, 20, 30, 42, 46], what type of decay should we expect when it is stacked to a membrane? Such a system is weakly coupled and falls within the general framework of the indirect stabilization of weakly coupled elastic systems, which has quite a rich literature, e.g. [1, 2, 3, 16, 18, 35, 41, 43, 45]. Unlike the works just cited dealing with the indirect stabilization of weakly coupled elastic systems, where one system is mechanically damped and the other one undamped, we are dealing here with a different type of indirect stabilization problem; more precisely, we are dealing with a doubly indirect stabilization problem in the sense that we are relying on the dissipation induced by the heat component of the system to strongly stabilize the coupled system. Given the weak coupling between the thermoelastic plate and the wave equations, uniform or exponential stability is not to be expected, thanks to a result of Triggiani [47]. Therefore, we will be focusing our attention on establishing a polynomial stability of the coupled system. Now, we shall introduce some notations, and formulate our problem. Let Ω ⊂ Rd, d ≥ 1, be an open bounded set of Rd with smooth enough boundary Γ. Let α and β be two nonzero real numbers with the same sign. Consider the coupled thermoelastic Kirchhoff plate/wave system ytt − γ∆ytt + a∆2y + α∆θ + µz = 0 in Ω× (0,+∞), θt − σ∆θ − β∆yt = 0 in Ω× (0,+∞), ztt − η∆z + µy = 0 in Ω× (0,+∞), y = ∂νy = 0, θ = z = 0, on Γ× (0,+∞), y(x, 0) = y0, yt(x, 0) = y1, θ(x, 0) = θ0 in Ω, z(x, 0) = z0, zt(x, 0) = z1 in Ω, (1.1) where a, η, γ, σ are positive physical constants representing respectively, the flexural stiffness of the plate, wave speed, rotational force constant, and thermal conductiv- ity, while µ denotes the coupling parameter, and is a nonzero real number. From a physical point of view, the parameters α, β and µ should be positive with α = β. Further, we assume that the coupling parameter µ satisfies |µ| < λ0µ0 √ aη, (1.2) where λ2 0 is the first eigenvalue of the operator “−∆” with Dirichlet boundary conditions, and µ2 0 is first eigenvalue of the operator ∆2 with clamped boundary conditions. This smallness condition on µ ensures that the right hand side of (1.3) below is positive for nonzero elements in the energy space, and thereby defines a norm indeed, which, thanks to Poincaré and Rellich inequalities, is equivalent to the natural norm in the energy space. We introduce the Hilbert space over the field C of complex numbers Hγ := H2 0 (Ω)×H1 0 (Ω)× L2(Ω)×H1 0 (Ω)× L2(Ω), EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 3 equipped with the norm ‖U‖2Hγ := a‖∆u‖2 + γ‖∇v‖2 + ‖v‖2 + α β ‖θ‖2 + η‖∇y‖2 + ‖z‖2 + 2µ ∫ Ω Re(uy)dΩ, (1.3) for all U = (u, v, θ, y, z) ∈ Hγ . We also define the linear differential operator AγU =  v −aP−1 γ ∆2u− αP−1 γ ∆θ − µP−1 γ y β∆v + σ∆θ z −µu+ η∆y  where Pγ = I − γ∆ which is an isomorphism of H1 0 (Ω) onto H−1(Ω) and U = (u, v, θ, y, z). The operator Aγ is unbounded in Hγ , and (thanks to elliptic regularity) its domain is D(Aγ) = [H3(Ω)∩H2 0 (Ω)]×H2 0 (Ω)× [H2(Ω)∩H1 0 (Ω)]× [H2(Ω)∩H1 0 (Ω)]×H1 0 (Ω). The system (1.1) can be recast as an abstract evolution system, U̇ = AγU, U(0) = U0 = (u0, v0, θ0, y0, z0). For the well-posedness of the system (1.1), we have the following result. Theorem 1.1. Operator Aγ is the infinitesimal generator of a C0-semigroup of contractions (Sγ(t))t≥0 on the Hilbert space Hγ . Proof. To prove that Aγ generates a C0-semigroup of contractions, we shall show that the conditions of the Lumer-Phillips theorem are satisfied [36, Theorem 4.3]; since the domain ofAγ is dense inHγ , we shall demonstrate here thatAγ is maximal dissipative. Let U = (u, v, θ, y, z) ∈ D(Aγ). We have (AγU,U) = a(∆v,∆u)L2(Ω) + (P 1/2 γ ( −aP−1 γ ∆2u− αP−1 γ ∆θ − µP−1 γ y ) , P 1/2 γ v)L2(Ω) + α β (β∆v + σ∆θ, θ)L2(Ω) + η(∇z,∇y)L2(Ω) + (−µu+ η∆y, z)L2(Ω) + µ ∫ Ω (vy + uz) dx = a(∆u,∆v)L2(Ω) − a ( ∆2u, v ) H−2(Ω),H2 0 (Ω) − α (∆θ, v)L2(Ω) − µ (y, v)L2(Ω) + α β (β∆v + σ∆θ, θ)L2(Ω) + η(∇z,∇y)L2(Ω) + (−µu+ η∆y, z)L2(Ω) + µ ∫ Ω (vy + uz) dx Using Green’s formula, we obtain Re(AγU,U) = −σα β ‖∇θ‖2 ≤ 0. Now let F = (f1, f2, f3, f4, f5) ∈ Hγ . We look for an element U = (u, v, θ, y, z) ∈ D(Aγ) such that (I −Aγ)U = F. 4 S. MANSOURI, L. TEBOU EJDE-2020/121 Equivalently, we consider the system u− v = f1, y − z = f4, (1.4) Pγu+ a∆2u+ α∆θ + µy = Pγ(f1 + f2), (1.5) θ − β∆u− σ∆θ = −β∆f1 + f3, (1.6) y − η∆y + µu = f4 + f5. (1.7) Taking φ ∈ H2 0 (Ω), ϕ ∈ H1 0 (Ω) and ψ ∈ H1 0 (Ω), multiplying (1.5) by φ, (1.6) by ϕ and (1.7) by ψ, we obtain the variational problem B((u, θ, y), (φ, ϕ, ψ)) = L((φ, ϕ, ψ)), where B((u, θ, y), (φ, ϕ, ψ)) = ∫ Ω { P 1/2 γ uP 1/2 γ φ+ a∆u∆φ− α∇θ∇φ+ α β θϕ + α∇u∇ϕ+ σ α β ∇θ∇ϕ+ yψ + η∇y∇ψ + µ(yφ+ uψ) } dx, and L((φ, ϕ, ψ)) = ∫ Ω { P 1/2 γ (f1 + f2)P 1/2 γ φ+ α β (−β∆f1 + f3)ϕ+ (f4 + f5)ψ } dx. We can easily check that B is a sesquilinear continuous and coercive map in [H2 0 (Ω)× L2(Ω) × H1 0 (Ω)]2 and L is a linear continuous form in H2 0 (Ω) × L2(Ω) × H1 0 (Ω). Thanks to Lax-Milgram Lemma, the above variational problem admits a unique solution (u, θ, y) ∈ H2 0 (Ω)×L2(Ω)×H1 0 (Ω), which shows that the operator I −Aγ is onto. � Now, we shall analyze the asymptotic behavior of system (1.1). Theorem 1.2. Let γ > 0. (1) The semigroup (Sγ(t))t≥0 is strongly stable on Hγ , lim t→+∞ ‖Sγ(t)U0‖Hγ = 0, ∀U0 ∈ Hγ . (2) The semigroup (Sγ(t))t≥0 is not exponentially stable. Proof. (1) To prove the strong stability of the semigroup, it suffices to check that the imaginary axis is included in the resolvent set, viz., iR ⊂ ρ(Aγ), where ρ(Aγ) is the resolvent set of Aγ . Before going forward, let us note that by the regularity theory for linear ellip- tic operators, if (u, v, θ, y, z) belongs to D(Aγ), then (u, v, θ, y, z) lies in H3(Ω) × H2(Ω)×H2(Ω)×H2(Ω)×H1 0 (Ω); so, in particular, the operator Aγ has a compact resolvent. Therefore, its spectrum is discrete. It is easy to check that 0 ∈ ρ(Aγ). Now we prove that iR\{0} ⊂ ρ(Aγ). Suppose that there exist λ ∈ R with λ 6= 0, and U = (u, v, θ, y, z) ∈ D(Aγ) with iλU −AγU = 0. (1.8) We shall prove that U = 0 = (0, 0, 0, 0, 0). Equivalently, we consider the system iλu− v = 0, iλy − z = 0, (1.9) iλPγv + a∆2u+ α∆θ + µy = 0, (1.10) iλθ − β∆v − σ∆θ = 0, (1.11) EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 5 iλz + µu− η∆y = 0. (1.12) Taking the inner product with U on both sides of (1.8), and taking the real parts, we immediately find θ = 0. Therefore (1.11) and the fact that v = 0 on Γ yield v = 0 in Ω. Then we obtain u = 0 by (1.9), since λ 6= 0. Next, we derive y = 0 from (1.10) and z = 0 by (1.9). Hence U = 0. Finally, Aγ has no purely imaginary eigenvalue, and so (Sγ(t))t≥0 is strongly stable, thanks to the semigroup strong stability criterion of Benchimol [10], or Arendt-Batty [5]. (2) We shall show that the semigroup (Sγ(t))t≥0 is not exponentially stable. We will use a result of Triggiani [47] on compact perturbations of semigroups. Let CµU =  0 µP−1 γ y 0 0 µu  for all U ∈ Hγ and A0 γ be the operator obtained from Aγ by setting µ = 0. Therefore, A0 γ = Aγ + Cµ. It is clear that A0 γ is a compact perturbation of Aγ . We consider a nonzero real number c and w ∈ H1 0 (Ω) such that −∆w = c2 η w. Let V =  0 0 0 w icw  . Then A0 γV = icV , so iR 6⊂ ρ(A0 γ), which shows that the semigroup generated by A0 γ is not strongly stable, hence not exponentially stable. Therefore, applying Triggiani’s result, we find that the semigroup (Sγ(t))t≥0 is not exponentially stable. � Theorem 1.3. The semigroup (Sγ(t))t≥0 is polynomially stable i.e. for all nonzero µ small enough, there exists C > 0 such that ‖Sγ(t)Z0‖γ ≤ C (1 + t) 1 6 ‖Z0‖D(Aγ), ∀t ≥ 0, ∀Z0 ∈ D(Aγ) Before proving the above theorem, we want to compare the polynomial estimate obtained here with the one established in [18, Theorem 3.1]. Remark 1.4. In [18, Section 3], the authors consider a mechanically damped Kirchhoff plate weakly coupled to an undamped wave equation, and prove that the corresponding semigroup satisfies for every positive integer m, γ > 0, and every nonzero α, there exists Cα,γ,m > 0 such that ‖Ŝα,γ(t)Z0‖α,γ ≤ Cα,γ,m‖Z0‖D(Âmα,γ) (1 + t) m 8 , ∀t ≥ 0, ∀Z0 ∈ D(Âmα,γ). Thus the decay of the semigroup in the case of a mechanically damped plate is O(t−1/8) when m = 1, while our polynomial stability result shows that, in the case of a thermoelastic plate, the decay rate of the semigroup is O(t− 1 6 ). Thus, our result shows that the decay of the semigroup in the case of a thermally damped plate is faster than in the case of a mechanically damped plate. We can also invoke 6 S. MANSOURI, L. TEBOU EJDE-2020/121 [9, Proposition 3.1], to derive from our theorem that, for every positive integer m, every γ > 0, and every nonzero constant µ, the semigroup satisfies the following decay estimate: there exists C > 0 such that ‖Sγ(t)Z0‖γ ≤ C‖Z0‖D(Amγ ) (1 + t)m/6 , ∀t ≥ 0, ∀Z0 ∈ D(Amγ ). Proof of Theorem 1.3. Thanks to a result of Borichev-Tomilov [12], it suffices to prove the resolvent estimate ‖(ibI −Aγ)−1‖L(Hγ) = O(|b|6), as |b| ↗ +∞. (1.13) To prove that resolvent estimate, we shall show that there exists C0 > 0 such that for every U ∈ Hγ , one has ‖(ibI −Aγ)−1U‖γ ≤ C0|b|6‖U‖γ , ∀b ∈ R, with |b| ≥ 1. Now, let U ∈ Hγ and let b a real number with |b| ≥ 1. There exists Z ∈ Aγ such that ibZ −AγZ = U. (1.14) We note Z = (u, v, θ, y, z), and U = (f, g, h, k, l). Taking the inner product with Z on both sides of (1.14), then taking the real parts, we immediately obtain |∇θ|22 ≤ C‖U‖γ‖Z‖γ , (1.15) where, hereafter, |q|2 stands for ‖q‖L2(Ω) and C denotes a generic positive constant that depends on the parameters of the system, but is independent of b. This constant varies from an inequality to another and it can vary even in the same line. With the notation above, equation (1.14) can be rewritten as ibu− v = f, (1.16) ibv + aP−1 γ ∆2u+ αP−1 γ ∆θ + µP−1 γ y = g, (1.17) ibθ − β∆v − σ∆θ = h, (1.18) iby − z = k, (1.19) ibz + µu− η∆y = l. (1.20) By applying the operator Pγ in equation (1.17), we obtain ibPγv + a∆2u+ α∆θ + µy = Pγg. (1.21) Now, multiplying (1.18) by v and integrating over Ω, we derive β|∇v|22 = ∫ Ω {−ibθ + h} v dx− ∫ Ω ∇θ∇v dx. (1.22) Using Cauchy-Schwarz inequality, Poincaré inequality, Young inequality and (1.15) yields |∇v|22 ≤ C(b2|θ|22 + |∇θ|2 + |h|2) ≤ C(b2‖U‖γ‖Z‖γ + ‖U‖γ‖Z‖γ + ‖U‖2γ) ≤ C(b2‖U‖γ‖Z‖γ + ‖U‖2γ). (1.23) Then by (1.16) and (1.23), we have b2|∇u|22 ≤ 2(|∇v|2 + |∇f |2) ≤ C(b2‖U‖γ‖Z‖γ + ‖U‖2γ). (1.24) and thanks Poincaré inequality, b2|u|22 ≤ C(b2‖U‖γ‖Z‖γ + ‖U‖2γ). (1.25) EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 7 Now we estimate each term in ‖Z‖γ . Estimation of |∆u|2: Substituting (1.16) in (1.21), we obtain − b2Pγu+ a∆2u+ α∆θ + µy = Pγg + ibPγf. (1.26) Multiplying (1.26) by u, integrating the resulting equation over Ω and using Green’s formula, we have |∆u|22 = b2 a |P 1/2 γ u|22 + α a ∫ Ω ∇θ∇u dx− µ a ∫ Ω yu dx + 1 a ∫ Ω {Pγg + ibPγf}u dx. (1.27) Using the Cauchy-Schwarz inequality, (1.15) and (1.24) we obtain |α a ∫ Ω ∇θ∇u dx| ≤ |α| a |∇θ|2|∇u|2 ≤ C ( ‖U‖γ‖Z‖γ + ‖U‖3/2γ ‖Z‖1/2γ ) . (1.28) By (1.19), we have |by|2 ≤ (|z|2 + |k|2) ≤ ( ‖Z‖γ + ‖U‖γ ) , (1.29) which, together with (1.25), yield | − µ a ∫ Ω yu dx| ≤ |µ| a |y|2|u|2 ≤ |µ| a |b|−2|by|2|bu|2 ≤ C ( b−1‖U‖1/2γ ‖Z‖3/2γ + b−2‖U‖γ‖Z‖γ + b−1‖U‖3/2γ ‖Z‖1/2γ + b−2‖U‖2γ ) , (1.30) and | ∫ Ω {Pγg + ibPγf}u dx| ≤ 1 2 ( |P 1/2 γ f |22 + |P 1/2 γ g|22 ) + b2|P 1/2 γ u|22 ≤ C‖U‖2γ + b2|P 1/2 γ u|22. (1.31) Now, using (1.28), (1.30) and (1.31) in (1.27), we have |∆u|22 ≤ C ( b2|P 1/2 γ u|22 + ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + b−1‖U‖1/2γ ‖Z‖3/2γ + ‖U‖2γ ) . (1.32) In the sequel we will use the estimate |∆u|22 ≤ C ( b2|P 1/2 γ u|22 + ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + ‖U‖2γ + |u|2|y|2 ) . (1.33) Estimation of b2|P 1/2 γ u|2. By (1.16), we have b2|P 1/2 γ u|22 ≤ 2 ( |P 1/2 γ v|22 + |P 1/2 γ f |22 ) . (1.34) Then it suffices to estimate |P 1/2 γ v|2. To this end, we use some multiplier techniques developed in [6, 42]. Multiply both sides of (1.18) by GPγv, where G = (−∆)−1 with −∆ considered with Dirichlet boundary conditions. Integrating over Ω and using Green’s formula, we derive ib ∫ Ω GθPγv dx+ β|P 1/2 γ v|22 + σ ∫ Ω P 1/2 γ θP 1/2 γ v dx = ∫ Ω P 1/2 γ (Gh)P 1/2 γ v dx. (1.35) 8 S. MANSOURI, L. TEBOU EJDE-2020/121 Thanks to Cauchy-Schwarz and Young inequalities, we have |P 1/2 γ v|2 ≤ | ib β ∫ Ω GθPγv dx|+ σ |β| |P 1/2 γ θ|2|P 1/2 γ v|2 + 1 |β| |P 1/2 γ (Gh)|2|P 1/2 γ v|2 ≤ | ib β ∫ Ω GθPγv dx|+ σ |β| |P 1/2 γ θ|22 + 1 |β| |P 1/2 γ (Gh)|22 + 1 2 |P 1/2 γ v|22. Then, by (1.15) we have |P 1/2 γ v|22 ≤ C ( |ib ∫ Ω GθPγv dx|+ ‖U‖γ‖Z‖γ + ‖U‖2γ ) . (1.36) It remains to estimate the first term in the right hand side of (1.36). Multiply (1.21) by Gθ and apply the Green’s formula to obtain ib ∫ Ω PγvGθ dx+ a ∫ Ω ∇u.∇θ dx− a ∫ Γ ∆u∂ν(Gθ) dΓ− α|θ|2 + µ ∫ Ω yGθ dx = ∫ Ω PγgGθ. Using the Cauchy-Schwarz inequality leads to the estimate∣∣ib∫ Ω PγvGθ dx ∣∣ ≤ C(|∇u|2|∇θ|2 + |∆u|L2(Γ)|∂νGθ|L2(Γ) + |α‖θ|22 + µ|y|2|Gθ|2 + |P 1/2 γ g|2|P 1/2 γ Gθ|2 ) . (1.37) Now, ∂ν ∈ L(H2(Ω), L2(Γ)), G ∈ L(L2(Ω), H2(Ω) ∩H1 0 (Ω)). Therefore, this fact, (1.15) and (1.24) yield∣∣ib∫ Ω PγvGθ dx ∣∣ ≤ C(‖U‖γ‖Z‖γ + ‖U‖ 3 2 γ ‖Z‖1/2γ ) + C|y|2|θ|2 + C|∆u|L2(Γ)|θ|2. (1.38) The combination of (1.36) and (1.38) leads to |P 1/2 γ v|2 ≤ C ( ‖U‖γ‖Z‖γ + ‖U‖ 3 2 γ ‖Z‖1/2γ + ‖U‖2γ ) + C|y|2|θ|2 + C|∆u|L2(Γ)|θ|2. (1.39) Estimation of |∆u|L2(Γ). Let ξ be a positive constant to be specified later. Let q ∈ [ C2(Ω) ]d be a vector field satisfying q = ν on Γ, see for example [22, 29]. Multiply (1.26) by ξu+ 2q.∇u and integrate the result over Ω to obtain − ξb2|P 1/2 γ u|22 − 2b2 Re ∫ Ω Pγuq · ∇u dx+ 2aRe ∫ Ω ∆2uq · ∇u dx + aξ|∆u|22 = Re ∫ Ω {−α∆θ − µy + Pγg + ibPγf}(ξu+ 2q · ∇u) dx. (1.40) Now, applying Green’s formula, we find 2 Re ∫ Ω ∆2u(q · ∇u) dx = − ∫ Ω div(q)|∆u|2 dx+ 2 Re ∫ Ω ∆qk ∂u ∂xk ∆u dx + 4 Re ∫ Ω ∇qk · ∇ ( ∂u ∂xk ) ∆u dx+ ∫ Γ q · ν|∆u|2dΓ EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 9 − 2 Re ∫ Γ ∆u∂ν(q · ∇u) dΓ + 2 Re ∫ Γ q · ∇u∂ν(∆u) dΓ. Thanks to the boundary conditions on u, one checks that ∂ν(q · ∇u) = q · ν∆u on Γ. Hence 2 Re ∫ Ω ∆2u(q · ∇u) dx = − ∫ Ω div(q)|∆u|2 dx+ 2 ∫ Ω ∆qk ∂u ∂xk ∆u dx + 4 ∫ Ω ∇qk · ∇ ( ∂u ∂xk ) ∆u dx− ∫ Γ |∆u|2 dΓ. (1.41) Proceeding similarly, we obtain Re ∫ Ω ∆u(2q · ∇u) dx = − ∫ Ω 2 Re(∇u · ∇(qk)∂ku+ qk∇u · ∇(∂ku)) dx+ 2 ∫ Γ |∂νu|2 dΓ = − ∫ Ω 2 Re(∇u · ∇(qk)∂ku) + qk∂k(|∇u|2) dx = − ∫ Ω 2 Re(∇u · ∇(qk)∂ku) dx+ ∫ Ω div(q)|∇u|2 dx, (1.42) and Re ∫ Ω u(2q · ∇u) dx = − ∫ Ω div(q)|u|2 dx, (1.43) which gives −2b2 Re ∫ Ω Pγuq · ∇u dx = −b2γ ∫ Ω 2 Re(∇u · ∇(qk)∂ku) dx + b2 ∫ Ω div(q)(|u|22 + γ|∇u|22) dx. (1.44) Reporting (1.41) and (1.44) in (1.40), we find a ∫ Γ |∆u|2dΓ + ξb2|P 1/2 γ u|22 − b2 ∫ Ω div(q)(P 1/2 γ u|2) dx + 2γb2 ∫ Ω Re(∇u · ∇(qk)∂ku) dx = ξa|∆u|22 − ∫ Ω div(q)|∆u|2 dx+ 2 ∫ Ω ∆qk ∂u ∂xk ∆u dx + 4 ∫ Ω ∇qk · ∇ ( ∂u ∂xk ) ∆u dx + Re ∫ Ω {α∆θ + µy − Pγg − ibPγf}(ξu+ 2q · ∇u) dx ≤ C|∆u|22 + Re ∫ Ω {α∆θ + µy − Pγg − ibPγf} (ξu+ 2q · ∇u) dx. (1.45) Now we estimate the last integral in the right hand side of (1.45). For that purpose, an application of Green’s formula yields∫ Ω ∆θ(ξu+ 2q.∇u) dx = 2 ∫ Ω ∇θ · ∇(q · ∇u) dx− ξ ∫ Ω ∇θ · ∇u dx 10 S. MANSOURI, L. TEBOU EJDE-2020/121 = 2 ∫ Ω ∇θ · ∇(qk)∂ku dx+ 2 ∫ Ω qk∇u · ∇(∂ku)) dx− ξ ∫ Ω ∇θ · ∇u dx. Applying the Cauchy-Schwarz inequality, we find that∣∣ ∫ Ω ∆θ(ξu+ 2q.∇u) dx ∣∣ ≤ C(|∇θ|2‖∇u|2 + |∇θ|2|∆u|2) ≤ C|∇θ|2|∆u|2 ≤ C(|∇θ|22 + |∆u|22) ≤ C‖U‖γ‖Z‖γ + C|∆u|22. (1.46) Similarly, and keeping in mind (1.25), we derive∣∣ ∫ Ω {µy − Pγg − ibPγf}(ξu+ 2q · ∇u) dx ∣∣ ≤ C(b−1|by|2|u|2 + b−1|by|2|∇u|2 + |P 1/2 γ g|2|∆u|2) + |b‖∆f |2|P 1/2 γ u|2 ≤ Cε ( |b|−1‖U‖1/2γ ‖Z‖3/2γ + ‖U‖γ‖Z‖γ + |b|−1‖U‖3/2γ ‖Z‖1/2γ + ‖U‖2γ ) + εb2|P 1/2 γ u|2 (1.47) Using (1.46) and (1.47) in (1.45) where ξ is chosen with ξ ≥ 2(‖ div(q)‖L∞(Ω) + 2‖∇q‖L∞(Ω)), we obtain∫ Γ |∆u|2dΓ + ξ 2 b2|P 1/2 γ u|22 ≤ εb2|P 1/2 γ u|22 + Cε ( |b|−1‖U‖1/2γ ‖Z‖3/2γ + ‖U‖γ‖Z‖γ + |b|−1‖U‖3/2γ ‖Z‖1/2γ + ‖U‖2γ ) + C|∆u|22 . Choosing ε = ξ/3, we obtain∫ Γ |∆u|2dΓ ≤ C ( |b|−1‖U‖1/2γ ‖Z‖3/2γ + ‖U‖γ‖Z‖γ + |b|−1‖U‖3/2γ ‖Z‖1/2γ + ‖U‖2γ ) + C|∆u|22. (1.48) Then by (1.15) and (1.48), the Poincaré inequality and the Young inequality, we have C|∆u|L2(Γ)|θ|2 ≤ C ( ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + ‖U‖3/4γ ‖Z‖5/4γ + ‖U‖5/4γ ‖Z‖3/4γ ) + |∆u|22 4 . (1.49) Finally from (1.33), (1.34), (1.39) and (1.49) we have |∆u|22 ≤ C ( ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + ‖U‖3/4γ ‖Z‖5/4γ + ‖U‖5/4γ ‖Z‖3/4γ + ‖U‖2γ ) + C(|u|2|y|2 + |y|2|θ|2). (1.50) Thanks (1.15) and (1.29), we have |y|2|θ|2 = b−1|by|2|θ|2 ≤ b−1 ( ‖U‖1/2γ ‖Z‖3/2γ + ‖U‖3/2γ ‖Z‖1/2γ ) . (1.51) EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 11 Using (1.51) and (1.30) in (1.50), we obtain |∆u|22 ≤ C ( ‖U‖2γ + ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖5/4γ ‖Z‖3/4γ + ‖U‖γ‖Z‖γ + ‖U‖3/4γ ‖Z‖5/4γ + |b|−1‖U‖1/2γ ‖Z‖3/2γ ) . (1.52) Estimation of |∇y|2. Reporting (1.19) in (1.20), multiplying the result by y and integrating over Ω, we obtain η|∇y|22 = b2|y|2 + Re ∫ Ω (ibk − µu+ l)y dx. (1.53) Using Hölder and Young inequalities and (1.25), we obtain∣∣ ∫ Ω (ibk − µu+ l)y dx ∣∣ ≤ |b‖k|2|y|2 + |µ‖u|2|y|2 + |l|2|y|2 ≤ b2|y|22 + C(|u|22 + |l|22 + |k|22) ≤ b2|y|22 + C(‖U‖γ‖Z‖γ + ‖U‖2γ) (1.54) Combining (1.53) and (1.54), we have |∇y|22 ≤ C(b2|y|22 + ‖U‖γ‖Z‖γ + ‖U‖2γ). (1.55) Estimation of b2|y|22: Multiplying (1.21) by 1 µy, integrating over Ω and using Green’s formula lead to |y|22 = − ib µ ∫ Ω Pγvy dx− a µ ∫ Ω ∆u∆y dx + a µ ∫ Γ ∆u∂νy dΓ + α µ ∫ Ω ∇θ · ∇y dx+ 1 µ ∫ Ω Pγgy dx. (1.56) Multiplying the conjugate of (1.20) by ∆u η and integrating over Ω, we derive∫ Ω ∆u∆y dx = 1 η ∫ Ω (−ibz + µu− l)∆u dx. (1.57) Therefore, b2|y|22 = b2 µ Re ∫ Ω (−ibPγv + Pγg)y dx+ ab2 µη Re ∫ Ω (ibz − µu+ l)∆u dx + ab2 µ Re ∫ Γ ∆u∂νy dΓ + αb2 µ Re ∫ Ω ∇θ · ∇y dx. (1.58) To simplify notations, we denote b2|y|22 = I1 + I2 + I3 + I4, (1.59) where Ii corresponds to the ith integral in the right-hand side in (1.58). Thus, we estimate each integral Ii. By the Poincaré inequality, we have |P 1/2 γ y|22 = |y|22 + γ|∇y|22 ≤ C|∇y|22. (1.60) 12 S. MANSOURI, L. TEBOU EJDE-2020/121 Then, using Cauchy-Schwarz inequality and (1.60), we have |I1| ≤ ∣∣b2 µ ∫ Ω −ibPγvy dx ∣∣+ ∣∣b2 µ ∫ Ω Pγgy dx ∣∣ ≤ ∣∣b2 µ ∫ Ω −ibPγvy dx ∣∣+ b2 |µ| |P 1/2 γ g|2|P 1/2 γ y|2 ≤ ∣∣b2 µ ∫ Ω −ibPγvy dx ∣∣+ Cb2‖U‖γ‖Z‖γ . (1.61) On the other hand,∫ Ω −ibPγvy dx = −ib ∫ Ω vy dx+ iγb ∫ Ω ∆vy dx. (1.62) By the Poincaré inequality, (1.23) and (1.29), we have ∣∣− ib∫ Ω vy dx ∣∣ ≤ |v|2|by|2 ≤ C|∇v|2|by|2 ≤ C ( |b|‖U‖1/2γ ‖Z‖3/2γ + |b|‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + ‖U‖2γ ) (1.63) Now multiplying (1.18) by y and integrating over Ω, we obtain∫ Ω ∆vy dx = 1 β ∫ Ω {ibθ − σ∆θ − h}y dx. (1.64) Thanks to Green’s formula in (1.64), Cauchy-Scwharz inequality and (1.29), we obtain |iγb ∫ Ω ∆vy dx| ≤ γ |β| ( b2|θ|2|y|2 + σ|b‖∇θ|2|∇y|2 + |b‖h|2|y|2 ) ≤ γ |β| (|b||θ|2|by|2 + σ|b‖∇θ|2|∇y|2 + |h|2|by|2) ≤ γ |β| ( (1 + σ)|b|‖U‖1/2γ ‖Z‖3/2γ + |b|‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + ‖U‖2γ ) (1.65) Combining (1.62), (1.63) and (1.65), we have ∣∣ ∫ Ω −ibPγvy dx ∣∣ ≤ C ( |b|‖U‖1/2γ ‖Z‖3/2γ + |b|‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + ‖U‖2γ ) (1.66) Using (1.66) in (1.61), we obtain |I1| ≤ C ( |b|3‖U‖1/2γ ‖Z‖3/2γ + |b|3‖U‖3/2γ ‖Z‖1/2γ + b2‖U‖γ‖Z‖γ + b2‖U‖2γ ) (1.67) EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 13 For the second integral I2 in (1.59), using Cauchy-Schwarz inequality and (1.25), we find |I2| ≤ | ab2 µη ∫ Ω ibz∆u dx|+ ab2 |µ|η (|u|2|∆u|2 + |l|2|∆u|2) ≤ ab2 |µ|η | ∫ Ω ibz∆u dx|+ a |µ|η ( |b||bu|2|∆u|2 + b2|l|2|∆u|2 ) ≤ ab2 |µ|η | ∫ Ω ibz∆u dx|+ Cb2 ( ‖U‖1/2γ ‖Z‖3/2γ + ‖U‖γ‖Z‖γ ) . (1.68) Now, using (1.16) in (1.18) and multiplying the resulting equation by z, we obtain∫ Ω ibz∆u dx = ∫ Ω { ib β θ + ∆f − 1 β h } z dx+ σ β ∫ Ω ∇θ∇z dx. (1.69) From (1.19), we have the estimate |∇z|2 ≤ √ 2(|b‖∇y|2 + |∇k|2). Then, using Cauchy-Schwarz inequality, (1.15) in (1.69), we obtain ∣∣ ∫ Ω ibz∆u dx ∣∣ ≤ ( |b| |β| |θ|2 + |∆f |2 + 1 |β| |h|2 ) |z|2 + √ 2σ |β| |∇θ|2(|b‖∇y|2 + |∇k|2) ≤ C ( |b|‖U‖1/2γ ‖Z‖3/2γ + ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ ) . (1.70) Combining (1.70) and (1.68), we have |I2| ≤ C ( |b|3‖U‖1/2γ ‖Z‖3/2γ + b2‖U‖3/2γ ‖Z‖1/2γ + b2‖U‖γ‖Z‖γ ) . (1.71) For the integral I4 in (1.59), using (1.15) and Young inequality, we have the estimate |I4| = ∣∣αb2 µ ∫ Ω ∇θ · ∇y dx ∣∣ ≤ Cb2|∇θ|2|∇y|2 ≤ Cb2‖U‖1/2γ ‖Z‖3/2γ . (1.72) Now it remains to estimate the boundary integral I3 in (1.59). To that end, we have to estimate |∂νy|L2(Γ) which can be estimated in the same way as [43, pp. 8-9]. Thus, we have |∂νy|L2(Γ) ≤ C ( |b‖y|2 + ‖U‖1/2γ ‖Z‖1/2γ + ‖U‖γ ) ≤ C ( ‖U‖1/2γ ‖Z‖1/2γ + ‖U‖γ + ‖Z‖γ ) . (1.73) Since by (1.48) and (1.52), we have |∆u|2L2(Γ) ≤ C ( ‖U‖2γ + ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖5/4γ ‖Z‖3/4γ + ‖U‖γ‖Z‖γ + ‖U‖3/4γ ‖Z‖5/4γ + |b|−1‖U‖1/2γ ‖Z‖3/2γ ) , (1.74) 14 S. MANSOURI, L. TEBOU EJDE-2020/121 it then follows from (1.73) and (1.74) that |I3| ≤ a |µ| b2|∆u|L2(Γ)|∂νy|L2(Γ) ≤ Cb2 ( (‖U‖3/2γ ‖Z‖1/2γ + ‖U‖γ‖Z‖γ + ‖U‖5/4γ ‖Z‖3/4γ + ‖U‖9/8γ ‖Z‖7/8γ + ‖U‖7/8γ ‖Z‖9/8γ ) + Cb2 ( ‖U‖7/4γ ‖Z‖1/4γ + ‖U‖13/8 γ ‖Z‖3/8γ + ‖U‖3/4γ ‖Z‖5/4γ + ‖U‖5/8γ ‖Z‖11/8 γ ) + Cb2 ( ‖U‖1/2γ ‖Z‖3/2γ + ‖U‖3/8γ ‖Z‖13/8 γ + |b|−1/2‖U‖1/4γ ‖Z‖7/4γ + ‖U‖2γ ) (1.75) Finally, reporting (1.63), (1.71), (1.72) and (1.75) in (1.59), we find b2|y|2 ≤ C|b|3 ( ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖1/2γ ‖Z‖3/2γ ) + Cb2 ( ‖U‖3/8γ ‖Z‖13/8 γ + ‖U‖2γ ) + Cb2 ( ‖U‖γ‖Z‖γ + ‖U‖5/4γ ‖Z‖3/4γ + ‖U‖9/8γ ‖Z‖7/8γ + ‖U‖7/8γ ‖Z‖9/8γ ) + Cb2 ( ‖U‖7/4γ ‖Z‖1/4γ + ‖U‖13/8 γ ‖Z‖3/8γ + ‖U‖3/4γ ‖Z‖5/4γ + ‖U‖5/8γ ‖Z‖11/8 γ ) + C|b|3/2‖U‖1/4γ ‖Z‖7/4γ . (1.76) Substituting (1.76) in (1.55) and combining (1.19) and (1.76), we find that |∇y|22 and |z|22 are bounded from above by the right hand side of (1.76). Since by the Cauchy-Schwarz inequality and (1.25), we have∣∣2µ∫ Ω Re(uy) dx ∣∣ ≤ 2|µ‖u|2|y|2 ≤ C ( ‖U‖1/2γ ‖Z‖3/2γ + |b|−1‖U‖γ‖Z‖γ ) (1.77) it follows from (1.15), (1.52), (1.76) and (1.77) that ‖Z‖2γ ≤ C|b|3 ( ‖U‖3/2γ ‖Z‖1/2γ + ‖U‖1/2γ ‖Z‖3/2γ ) + Cb2 ( ‖U‖3/8γ ‖Z‖13/8 γ + ‖U‖2γ ) + Cb2 ( ‖U‖γ‖Z‖γ + ‖U‖5/4γ ‖Z‖3/4γ + ‖U‖9/8γ ‖Z‖7/8γ + ‖U‖7/8γ ‖Z‖9/8γ ) + Cb2 ( ‖U‖7/4γ ‖Z‖1/4γ + ‖U‖13/8 γ ‖Z‖3/8γ + ‖U‖3/4γ ‖Z‖5/4γ + ‖U‖5/8γ ‖Z‖11/8 γ ) + C|b|3/2‖U‖1/4γ ‖Z‖7/4γ . Now, applying Young inequality several times and successively, one derives ‖Z‖2γ ≤ Cb12‖U‖2γ . Hence ‖(ibI −Aγ)−1U‖γ ≤ Cb6‖U‖γ , ∀U ∈ Hγ , ∀b ∈ R, |b| ≥ 1, By applying the Borichev-Tomilov result [12, Theorem 2.4], we complete the proof. � EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 15 References [1] F. Alabau-Boussouira; Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541. [2] F. Alabau, P. Cannarsa, V. Komornik; Indirect internal stabilization of weakly coupled sys- tems, J. Evolution Equations, 2 (2002), 127-150. [3] F. Alabau-Boussouira, P. Cannarsa, R. Guglielmi; Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Math. Control Relat. Fields 1 (2011), 413-436. [4] F. Ammar Khodja, A. Benabdallah, D. Teniou; Dynamical Stabilizers and Coupled Systems. ESAIM: Proceedings, 2 (1997), 253-262. [5] W. Arendt, C. J. K. Batty; Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc., 306 (1988), 837-852. [6] G. Avalos, I. Lasiecka; Exponential stability of a thermoelastic system without mechanical dissipation, Dedicated to the memory of Pierre Grisvard. Rend. Istit. Mat. Univ. Trieste, 28 (1996), suppl., 1-28 (1997). [7] George Avalos, Irena Lasiecka; Exponential stability of an uncontrolled thermoelastic system with varying boundary conditions. Appl. Anal. 68 (1998), 31-49. [8] G. Avalos, I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182. [9] A. Bátkai, K.-J. Engel, J. Prüss, R. Schnaubelt; Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. [10] C. D. Benchimol; A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization, 16 (1978), 373-379. [11] E. Bisognin, V. Bisognin, G. Perla Menzala, E. Zuazua; On exponential stability for von Kármán equations in the presence of thermal effects. Math. Methods Appl. Sci., 21 (1998), 393-416. [12] A. Borichev, Y. Tomilov; Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. [13] M. M. Cavalcanti, V. N. Domingos Cavalcanti, L. Tebou; Stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanisms. Electron. J. Differential Equations, 2017 (2017), No. 83, 18 pp. [14] C. Dafermos; On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271. [15] F. Dell’Oro, J. E. Muñoz Rivera, V. Pata; Stability properties of an abstract system with applications to linear thermoelastic plates. J. Evol. Equ., 13 (2013), 777-794. [16] X. Fu; Sharp decay rates for the weakly coupled hyperbolic system with one internal damping. SIAM J. Control Optim. 50 (2012), 1643-1660. [17] J. S. Gibson; A note on stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316. [18] A. Hajej, Z. Hajjej, L. Tebou; Indirect stabilization of weakly coupled Kirchhoff plate and wave equations with frictional damping, J. Math. Anal. Appl., 474 (2019), 290-308. [19] F. L. Huang; Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations,1 (1985), 43-56. [20] V. Keyantuo, L. Tebou, M. Warma; A Gevrey class semigroup for a Thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Dis- crete Contin. Dyn. Syst. A, 40 (2020), 2875-2889. [21] J. U. Kim; On the energy decay of a linear thermoelastic bar and plate. SIAM J. Math. Anal., 23 (1992), 889-899. [22] V. Komornik; Exact controllability and stabilization. The multiplier method, RAM, Masson & John Wiley, Paris, 1994. [23] J. E. Lagnese; Boundary Stabilization of Thin Plates. SIAM, Philadelphia (1989). [24] I. Lasiecka, R. Triggiani; Analyticity of thermo-elastic semigroups with free boundary condi- tions. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 27 (4) (1998), 457-482. [25] I.Lasiecka, R. Triggiani; Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations. Adv. Differential Equations, 3 (1998), 387-416. [26] I.Lasiecka, R. Triggiani; Analyticity and lack thereof, of thermo-elastic semigroups. Con- trol and partial differential equations (Marseille-Luminy, 1997), 199-222 (electronic), ESAIM Proc., 4, Soc. Math. Appl. Indust., Paris, 1998. 16 S. MANSOURI, L. TEBOU EJDE-2020/121 [27] Irena Lasiecka, Roberto Triggiani; Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C., Abstr. Appl. Anal., 3 (1998), 153-169. [28] G. Lebeau, E. Zuazua; Decay rates for the three-dimensional linear system of thermoelastic- ity. Arch. Ration. Mech. Anal., 148 (1999), 179-231. [29] J.-L. Lions; Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués, Vol. 1, RMA 8, Masson, Paris, 1988. [30] K. Liu, Z. Liu; Exponential stability and analyticity of abstract linear thermoelastic systems. Z. Angew. Math. Phys., 48 (1997), 885-904. [31] Z. Liu, B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56(2005), 630-644. [32] Z. Liu, M. Renardy; A note on the equations of thermoelastic plate. Appl. Math. Lett., 8 (1995), 1-6. [33] J. E. Muñoz Rivera, H. Portillo Oquendo; A transmission problem for thermoelastic plates. Quart. Appl. Math., 62 (2004), 273-293. [34] J. E. Muñoz Rivera, R. Racke; Smoothing properties, decay and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547- 1563. [35] H. P. Oquendo, R. P. Raya; Best rates of decay for coupled waves with different propagation speeds. Z. Angew. Math. Phys., 68 (2017), no. 4, Art. 77, 8 pp. [36] A. Pazy; Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [37] G. Perla-Menzala, E. Zuazua; The energy decay rate for the modified von Kármán system of thermoelastic plates: An improvement. Applied Mathematics Letters, 16 (2003), 531-534. [38] J. Prüss; On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. [39] J. Rauch, X. Zhang, E. Zuazua; Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84 (9), (2005), 407-470. [40] D. L. Russell; Decay rates for weakly damped systems in Hilbert space obtained with control- theoretic methods. J. Differential Equations, 19 (1975), 344-370. [41] D. L. Russell; A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358. [42] L. Tebou; Stabilization of some coupled hyperbolic/parabolic equations, Discrete Contin. Dyn. Syst. B, 14 (2010), 1601-1620. [43] L. Tebou; Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equa- tions with indirect damping mechanisms, Math. Control Relat. Fields, 2 (2012), 45-60. [44] L. Tebou; Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions. C. R. Math. Acad. Sci. Paris 351 (2013), 539-544. [45] L. Tebou; Indirect stabilization of a Mindlin-Timoshenko plate. J. Math. Anal. Appl., 449 (2017), 1880-1891. [46] L. Tebou; Can the average temperature stabilize a system of thermoelastic plates?, Vietnam Journal of Mathematics. To appear. [47] R. Triggiani; Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. [48] X. Zhang, E. Zuazua; Decay of solutions of the system of thermoelasticity of type III. Com- mun. Contemp. Math., 5 (2003), 25-83. Sabeur Mansouri Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia Email address: m.sabeur1@gmail.com Louis Tebou Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA Email address: teboul@fiu.edu 1. Introduction References