STABILITY FOR THERMOELASTIC PLATES WITH TWO TEMPERATURES

. We investigate the well-posedness, the exponential stability, or the lack thereof, of thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoﬀ type plates, two temperatures are involved, related by an elliptic equation. The arising initial boundary value problems for diﬀerent boundary conditions deal with systems of partial diﬀerential equations involving Schr¨odinger like equations, hyperbolic and elliptic equations, which have a diﬀerent character compared to the classical one with the usual single temperature. Depending on the model – with Fourier or with Cattaneo type heat conduction – we obtain exponential resp. non-exponential stability, thus providing another examples where the change from Fourier’s to Cattaneo’s law leads to a loss of exponential stability.


Introduction. Thermoelastic plates of Kirchhoff type modeled by
for (u, θ, q) = (u, θ, q)(t, x) denoting the displacement, the temperature and heat flux in a smoothly bounded domain Ω ⊂ R n , t ≥ 0, x ∈ Ω, with b, d, κ > 0, τ ≥ 0, have been discussed in recent years with respect to well-posedness and asymptotic behavior in time (also for unbounded domains) and both for τ = 0 and for τ > 0. So-called non-simple materials are modeled by two temperatures, the thermodynamic temperature θ and the conductive temperature ϕ, related to each other in 6334 RAMÓN QUINTANILLA AND REINHARD RACKE the following way, see [2,3,4,32], with a constant a ≥ 0. The corresponding extension of the classical thermoelastic plate model (1)-(3) then reads as τ q t + q + κ∇ϕ = 0, θ − ϕ + a∆ϕ = 0.
Boundary conditions will be given for u, ϕ and/or q below. For a = 0 we recover (1)- (3). In this case, in particular, it is known that one has, for appropriate boundary conditions, exponential stability for the Fourier type heat conduction given by τ = 0, while it is not exponentially stable for the Cattaneo (Maxwell) type given for a positive relaxation constant τ > 0, see [26,27,8]. Also the Cauchy problem (Ω = R n ) has been investigated in [30], where the loss of exponential stability in bounded domains is reflected in a regularity loss in the description of polynomial decay of solutions.
Here, we shall investigate initial boundary value problems for the case a > 0. We are first interested in the well-posedness both for τ = 0 and for τ > 0, which is more delicate in comparison to the case a = 0, since there will be no regularity gain in the temperature triggered by the differential equations. The second main topic will be to investigate exponential stability for τ = 0 resp. non-exponential stability for τ > 0. In particular for the case τ = 0, we will consider several boundary conditions, using different methods.
This way we also contribute a further example where the different heat conduction models, one by Fourier (τ = 0), one by Cattaneo (τ > 0) predict different qualitative behavior, see [9] for the thermoelastic Timoshenko model, and [26,27,8] for the classical thermoelastic model (a = 0). As was indicated in [29], these might not be exceptions, that is, the change from Fourier's to Cattaneo's "law" is likely to lead to a loss of exponential stability.
Case a > 0. Here only the second-order system in one space dimension in a bounded interval has been studied with respect to exponential stability for a = 0 [25], the well-posedness was obtained in any space dimension [24]. The non-exponential stability for τ > 0 was proved in in [18]. We carry over considerations to the fourth-order thermoelastic plate, which exhibits more complex difficulties, cp., for example, Section 7.

Our main new contributions are
• First discussion of the fourth-order thermoelastic plate system with two temperatures. • Proof of well-posedness for rather weak regular solutions, both for τ = 0 and for τ > 0. • Proof of exponential stability (τ = 0) resp. the lack thereof (τ > 0) for different boundary conditions with different methods; this way also providing another example for the problem of "right" modeling with Fourier or Cattaneo type laws. • Providing the rigorous proof of exponential stability for the second-order system.
The paper is organized as follows. In Section 2, the well-posedness for the thermoelastic plate model (5)-(8) (with appropriate boundary conditions and initial conditions) is investigated for τ = 0, while the corresponding results for τ > 0 are given in Section 3. The exponential stability for (5)-(8) is shown for τ = 0 for the boundary conditions (17) and (19) in Section 4 with semigroup methods, while Section 5 shows the general non-exponential stability for τ > 0. The exponential stability for the second-order thermoelastic model (9)-(12) is proved for τ = 0 in Section 6. The exponential stability for (5)-(8) is proved for τ = 0 in Section 7 for Robin type boundary conditions on the temperature with energy and lifting methods.
We use standard notation, in particular the Sobolev spaces L p = L p (Ω), p ≥ 1, and H s = W s,2 (Ω), s ∈ N 0 , with their associated norms · L p resp. · H s . The inner product in a Hilbert space X is given by ·, · X , and ·, · := ·, · L 2 . By Id we denote the identity on some given space.

2.
Well-posedness for τ = 0. Here we extend the work on the second-order system in [25] to the fourth-order system.
We start proving the well-posedness of the system (5)-(8) with τ = 0, i.e. for recalling that a > 0. Initial conditions are given by while boundary conditions are prescribed either by or by or by where ν denotes the exterior normal at the boundary. We write
which can be obtained by integrating (22).
Remark 1. The equation (22) for, essentially, θ does not trigger any regularity for θ, in contrast to the situation where a = 0 (only one temperature θ = ϕ). For a = 0 we would have the classical operator B = κ∆ on its usual domain. On the other hand, in equation (21) one needs, yet formally, ∆θ. This lack of regularity will be reflected in a lack of separate for regulariy for u and θ. We shall have a connected regularity, see below.
The operator B satisfies for θ = ϕ − a∆ϕ We transform the system (21), (22) into a system of first order in time for V := (u, u t , θ) , where denotes the transposed matrix: This formal system with the formal differential symbol A f will be considered as an evolution equation in an associated Hilbert space with inner product V,Ṽ H := b ∆u, ∆ũ + v,ṽ + θ,θ .
where (19). (29) In the definition of D(A), the problem of the missing (separate) regularity for θ is reflected. cp. Remark 1. One just has the combined regularity ∆(b∆u + dθ) ∈ L 2 , not writing ∆ 2 u, ∆θ ∈ L 2 , and this way A f V has to be interpreted.
As usual, ∆(b∆u + dθ) ∈ L 2 means We will show that A generates a contraction semigroup.
Consider the sesquilinear form β : K → C, where (17), and the associated variational problem to find, for any (h, g) ∈ L 2 × L 2 ( * ) , a unique (u, θ) ∈ K satisfying Solving (32) with gives the solution (u, θ) to (31). Here we use for the case of the boundary conditions (19) the self-adjointness of the Dirichlet-Laplace operator in The solvability of the variational problem follows from the Theorem of Lax and Milgram, observing with some positive constant c > 0, and, using the boundary conditions, with some positive constant c > 0 by elliptic regularity.
By the Lumer-Phillips theorem we conclude Theorem 2.3. A generates a contraction semigroup, and, for any V 0 ∈ D(A), there is a unique solution V to (27) satisfying By the contractivity we obtain the following stability estimate for the solution V to (27), for t ≥ 0, with a constant k > 0 not depending on t or on the data. More precisely, we have for the associated energy the relation for any V 0 ∈ D(A) and t ≥ 0. This can be easily seen multiplying (21) by u t (in L 2 ), and (22) by θ, adding, and using u t (t, ·) ∈ H 2 0 .
3. Well-posedness for τ > 0. The model (5)-(8) for thermoelastic plates of Kirchhoff type with two temperatures under the Cattaneo law, i.e. for τ > 0, will be shown to be well-posed for two different boundary conditions.. The well-posedness requires the choice of suitable representations of the solutions and corresponding phase spaces. The regularity issue is even more complicated due to the fact that the heat flux is not immediately of the same regularity as the gradient of the temperature ϕ, as it was in the case of the Fourier model discussed in the previous section.
The issue of only combined regularity for (u, θ, q) only, in contrast to separate regularity for each of u, θ, q, comes up again requiring the right spaces and domains of operators. Initial conditions are given by while boundary conditions are prescribed either as before in (17) by or as before in (19) by Defining w := u t , we obtain from (35)-(38) where B = κ∆ (Id − a∆) −1 denotes the operator defined in (23). Solving (42), (43) with inital conditions and boundary conditions on ∂Ω, either or one gets a solution (u, θ, q) to (35)-(40) by integration. For the solvability for (w, θ), we transform it to a first-order system W := (w, w t , θ, θ t ) , System (47) together with the boundary conditions (45) or (46) will be considered as an evolution equation in the associated Hilbert space with inner product V,Ṽ H := b ∆w, ∆w + z,z + θ,θ + y,ỹ .
In the definition of D(A 1 ), the problem of the missing (separate) regularity for ∆w, y is reflected again, cp. Section 2.
As in Section 2 one can show that A 1 generates a C 0 -semigroup. For this purpose we write The operator is bounded, and for (ii) The range of Id − A 11 equals H 1 .
As a consequence we obtain the well-posedness of (49), corresponding to the boundary conditions (45) or (46), Theorem 3.2. A 1 generates a C 0 -semigroup, and, for any W 0 ∈ D(A 1 ), there is a unique solution W to (49) satisfying In Section 5 we will demonstrate for the hinged boundary conditions (46) that there is not exponential decay. That is, we will have another example of a thermoelastic system for which the modeling with he Fourier law leads to an exponentially stable system (see Section 4), while the modeling with the Cattaneo law does not yield exponential stability (see Section 5). 4. Exponential stability for τ = 0. Here we will show the exponential stability of the thermoelastic plate with two temperatures for τ = 0, and for both boundary conditions (17) or (19) using semigroup methods. We use the following characterization of exponential stability given in [20] going back to Gearhart [10], Huang [11] and Prüß [23].
The three steps to obtain exponential stability for {e tA } t≥0 , with A from Section 2, are: Step 1: Prove 0 ∈ (A).
With the notation W n = (u n , v n , θ n ) and F n = (F 1 n , F 2 n , F 3 n ) we conclude iλ n u n − v n = F 1 n , iλ n θ n − Bθ n − d∆v n = F 3 n .
Finally, to prove Step 3 we can argue by assuming that (52) does not hold and conclude a contradiction in the same way as in Step 2, the only difference being that |λ n | → ∞ instead of λ n → ω.
Altogether we have proved the exponential stability.
Theorem 4.2. Solutions to the thermoelastic plate equation with two temperatures, both for the boundary conditions (17) and for the boundary conditions (19), tend to zero exponentially uniformly, i.e. the associated semigroup {e tA } t≥0 is exponentially stable.
We will prove the existence of slowly decaying solutions giving the non-exponential stability. That is we have another example where Fourier models gives exponential stability (according to Section 4; cp. also Section 6), while the Cattaneo model is not exponentially stable.
Proof. Let (χ j ) j denote the eigenfunctions of the Laplace operator for Dirichlet boundary conditions, −∆χ j = λ j χ j , χ j = 0 on ∂Ω, with eigenvalues 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ j → ∞ as j → ∞. We make the ansatz . This ansatz is compatible with the differential equations (61)-(64) and with the boundary conditions (65). It gives a solution (u j , q j , ϕ j , θ j ) if the coefficients (a j , b j , d j ) satisfy the following system of ODEs, where a prime denotes here differentiation with respect to time t, System (66) is equivalent to a first-order system for the column vector V j := (a j , a j , b j , q j ), We are looking for solutions to (67) of type V j (t) = e ωj t V 0 j . In other words, ω j has to be an eigenvalue of A j with eigenvector V 0 j as initial data. It is the aim to demonstrate that, for any given small ε > 0, we have some j and some eigenvalue ω j such that the real part Re ω j of ω j is larger than −ε. This will contradict the exponential stability. We have To show that ∀ ε > 0 ∃ j ∃ ω j , P j (ω j ) = 0 : Re ω j ≥ −ε, we introduce, for small ε > 0, That is, we have to show To prove (68) we start with computing P j,ε (z) = q 4 z 4 + q 3 z 3 + q 2 z 2 + q 1 z + q 0 where q 4 = τ (1 + aλ j ),

RAMÓN QUINTANILLA AND REINHARD RACKE
The coefficients q 0 , . . . , q 4 are positive for sufficiently small ε and large j, since λ j → ∞. We use the Hurwitz criterion [31]: Let denote the Hurwitz matrix associated to the polynomial P j,ε . Then (68) is fulfilled if we find, for given small ε > 0, a (sufficiently large) index j such that one of the principal minors of H j is not positive. The principal minors are given by the determinants det D j m of the matrices D j m , for m = 1, 2, 3, 4, where D j m denotes the upper left square submatrix of H j consisting of the elements H j 11 , . . . , H j mm . We compute 6. Exponential stability for the second-order system. The second-order thermoelastic system with two temperatures (9)-(12) is for τ = 0 given by together with initial conditions, and boundary conditions we avoid trivial non-decaying solutions as (u ≡ 0, θ ≡ 1). We notice that by equation (70) and the boundary conditions (74) the property of mean value zero for θ is invariant in time. i.e. assuming (75), we have for any t ≥ 0 In [24] Quintanilla investigated time-harmonic solutions showing that the real parts of the possible frequencies are strictly less than zero indicating exponential stability of the semigroup. Here we strictly prove the exponential stability by energy estimates, on one hand in order to complete [24], and on the other hand, as first demonstration of possible considerations for proving exponential stability with energy methods in the case of two temperatures. The latter will be considered for Robin type boundary conditions in the temperature for the fourth-order system in Section 7.
The well-posedness of (69)-(74) has been studied (for Dirichlet type boundary conditions instead of Neumann type boundary conditions for ϕ) by Quintanilla in [25] for any space dimension n = 1, 2, 3.
As regularity of the solutions we have we rewrite (69)-(72) as Defining the associated energy we have, cp. (34), (24), where c 1 will denote a positive constant in this section, possibly varying from line to line, and not depending on t or the initial data. Multiplying (78) by u in L 2 , we obtain d dt Because of the given regularity, not providing higher regularity for u and θ separately, we cannot use some multipliers typically used in thermoelasticity, cp. [28]. Therefore, we lift the regularity by integrating in space. For this purpose let Integrating (79) we get η(t, ·) ∈ H 1 0 and

RAMÓN QUINTANILLA AND REINHARD RACKE
Let P denote the operator given by Remark 2. Let B denote the operator defined in (23) in Section 2 again, now for Ω = (0, 1) Then we have This is equivalent to proving p = q x for p and q satisfying which easily follows using the boundary conditions and uniqueness for the boundary value problems.
Multiplying (83) by u t , we obtain for some (generic) c > 0, we conclude from (84) Combining (81) and (85) we get Defining the Lyapunov functional L 1 for ε > 0 by and choosing ε small enough, we obtain from (80) and (86) with some α 1 > 0 depending on ε and on c 1 (i.e. the coefficients of the differential equations). Observing for sufficiently small ε, we have, for any t ≥ 0, and then, by (76), and, finally, obtain Theorem 6.1. The energy E 1 for the system (78), (79) decays exponentially. For any t ≥ 0 we have with a positive constant α which does not depend on t or on the data. In terms of the associated semigroup, we have exponential stability.
The statement on the exponential stabilty of the semigroup follows, as usual, from the fact, that after transforming to a first-order system for V = (u, u t , θ) as in Section 2, the square of the norm of the solution is equivalent to E 1 .
The discussion in this section prepares the one of the next section for the fourthorder operator where the (low) regularity of the solution also will require a (more complicated) lifting in space, here reflected by the definition of η above.
7. Exponential stability for the fourth-order system for τ = 0 for Robin type boundary conditions. We prove the exponential stability for the thermoelastic plate equation with two temperatures and τ = 0, (13)- (16), (18), in one space dimension, with boundary conditions, for t ≥ 0, (95) These mixed periodic-Robin-type boundary conditions for ϕ are different from those considered in Section 2, so we will first give the arguments for the existence of solutions. The reason for considering these boundary conditions is mainly technical: the Dirichlet resp. Neumann boundary conditions for ϕ, studied in (17) resp. (18) in Section 2 and discussed in Section 4 in any space dimention, do not allow to carry over the energy approach below. The choice of the function h below has to assure h ∈ H 2 0 and simultaneously to be compatible with different aspects of the equations. Lifting the regularity, expressed in the definition of h and corresponding to the lifting given for the second-order problem in the previous section in the definition of η in (82), seems appropriate.
For the existence of solutions in appropriate space we first consider the operator Proof. By the theory of boundary value problems for ODEs, it is sufficient to check that the homogeneous problem only has the trivial solution. Let ϕ solve (96). Then Equation (97) is never satisfied for 0 < a < ∞ unless α = 0 which implies ϕ = 0. The argument for this is, for example, to look, for γ := 1 √ a ∈ (0, ∞), at the function f (γ) := (2 − γ) e γ + (2 + γ) e −γ − 4.
We rewrite the differential equations as in Section 2 in the form where The transformation to a first-order system for V := (u, u t , θ) reads as where The unique solvability is obtained as in Section 2 giving Theorem 7.2. A m generates a contraction semigroup, and, for any V 0 ∈ D(A m ), there is a unique solution V to (101) satisfying The exponential stability is of course only on the orthogonal complement of the null space of A m .
For the following Lemma, the special boundary conditions for ϕ are important. x (dv xx (x) + B m θ(x)) dx = 0.
Since v ∈ H 2 0 , we have Defining the associated energy where c 1 will denote a positive constant in this section, possibly varying from line to line, and not depending on t or the initial data. (102) follows, using R 1 [ϕ] = R 2 [ϕ] = 0, from ≤ −κa ϕ xx 2 L 2 . Multiplying (98) by u in L 2 , we obtain d dt The regularity lifting is done as follows. Let That is, the reason for looking at the special boundary conditions was to, finally, achieve h ∈ H 2 0 , which will allow a kind of partial integration in (107). Integrating (99) twice, using u ∈ H 2 0 , we obtain h t − P 1 B m θ − du t = 0, where (P 1 w)(x) := x 0 y 0 w(z)dzdy defines a bounded operator in L 2 . Multiplying (106) by u t in L 2 , we obtain The last equality was obtained because h ∈ H 2 0 . Thus with some (generic) constant c > 0. Combining (103) and (108) we get d dt Defining the Lyapunov functional L m for ε > 0 by Theorem 7.6. The energy E m for the system (98), (99) decays exponentially. For any t ≥ 0 we have E m (t) ≤ 3E m (0)e −αt . with a positive constant α which does not depend on t or on the data. In terms of the associated semigroup, we have exponential stability.
The statement on the exponential stability of the semigroup follows again from the fact, that after transforming to a first-order system for V = (u, u t , θ) , the square of the norm of the solution is equivalent to E m .