Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

This paper is concerned with long-time dynamics of a full von Karman system subject to nonlinear thermal coupling and free boundary conditions. In contrast with scalar von Karman system, vectorial full von Karman system accounts for both vertical and in plane displacements. The corresponding PDE is of critical interest in flow structure interactions where nonlinear plate/shell dynamics interacts with perturbed flows [vicid or invicid] [ 8 , 9 , 15 ]. In this paper it is shown that the system admits a global attractor which is also smooth and of finite fractal dimension. The above result is shown to hold for plates without regularizing effects of rotational inertia and without any mechanical dissipation imposed on vertical displacements. This is in contrast with the literature on this topic [ 15 ] and references therein. In order to handle highly supercritical nature of the von Karman nonlinearities, new results on "hidden" trace regularity generated by thermal effects are exploited. These lead to asymptotic compensated compactness of trajectories which then allows to use newly developed tools pertaining to quasi stable dynamical systems [ 8 ].


1.
Introduction. This paper is concerned with long-time behavior and theory of global attractors associated with dynamic system of nonlinear elasticity modeled by a full vectorial von Karman system subject to thermal effects. This system describes nonlinear oscillations in a plate dynamics which account for both vertical and in plane displacements -denoted respectively by w and u = (u 1 , u 2 ) -as well as the averaged thermal stresses φ and θ affecting each of these displacements [17,18,34,35]. The introduced mathematical model, being of interest on its own, is also a prototype for shallow shells with thermal effects [46,52]. The latter are the building blocks for flow-structure interactions which have attracted a considerable attention in recent literature [9,11,15,16]. In fact, the importance and interest in studying dynamical properties of vectorial Karman system can not be overstated. This is particularly pronounced for plate/shell models without the regularizing effects of rotational inertia. Indeed, stabilizing effect of the flow can only be attested for very thin plates which do not account for rotational inertia [9] and references therein. It is thus critical to be able to handle the analysis of full vectorial von Karman system without rotational inertia. On the other hand, mathematical treatment of such models is challenging due to severe singularities caused by nonlinear effects which are no longer mitigated by the additional regularity of vertical velocity exhibited in rotary inertial models. We shall exploit thermal effects as the carriers and propagators of partial regularity which, in turn, will allow for construction of a well posed dynamical system with a smooth and finite-dimensional long time behavior.
The goal of the paper is to establish existence of global attractor which captures asymptotic behavior of the nonlinear dynamics. In addition, we shall prove that such attractor is both finite dimensional and smooth. We note that the model neither includes mechanical dissipation on vertical displacements of the plate, nor accounts for rotational inertia term γ∆w tt which has regularizing effect on the dynamics. This is in striking contrast with the most of the literature on the topic [5,15,32,33,34,35,37]. In fact, since both regularity of the dynamics and a presence of sufficient dissipation have critical bearing on establishing smooth asymptotic behavior of the trajectories, proving such property for a model which has only limited dissipation and limited regularity is the main challenge undertaken in the present paper. Even more, γ > 0 is essential in proving uniqueness of weak solutions to a full vectorial von Karman system [28]. With γ = 0 the uniqueness and wellposedness of the corresponding dynamical system must be harvested from thermal effects.
Thus we are dealing with a loss of 1 + ε derivative. This feature becomes a major difficulty in the study of Hadamard wellposednesss (uniqueness and continuous dependence on the initial data) and, above all, in obtaining the needed estimates for the existence of attractors. While parabolic like structure is typically equipped with additional regularity properties, the challenge in the present problem is the "transfer" of these beneficial effects to the hyperbolic part of the system. The carriers of propagation in the case of free boundary conditions are boundary traces. Thus, at the technical level, we will be concerned with "hidden" trace regularity properties which will play a role of propagators of both regularity and stability. It is well known that the analysis of free boundary conditions, in the context of thermoelasticity, is a challenging subject-even within the linear theory [41]. This is due to the fact that Kirchhoff plate with free boundary conditions does not satisfy strong Lopatinski condition [50]. It is well known that Lopatinski condition is responsible for "hidden" regularity of boundary traces in hyperbolic dynamics [40,50]. In the absence of such, other tools based on microlocal calculus need to be brought to the analysis.
To our best knowledge the present paper is a first study of attractors for the dynamics described by full vectorial von Karman thermoelastic system with free boundary conditions and with no dissipation, nor regularity imposed on vertical displacement. This will be even more evident from the detailed review of the literature provided below.
1.2. Discussion of the literature. The analysis of wellposedness and of long-time behavior in nonlinear thermoelasticity has been a subject of long lasting interest [1,19,32,45,46]. Various models with different boundary conditions have been considered. However the physical interest-relevance and the degree of mathematical challenge does depend critically on the specific model and the associated boundary conditions. These create different configurations that require diverse mathematical treatments. The overriding desire has been to control long-time behavior of the model with a minimal amount of dissipation. By controlling, we mean either to steer trajectories to zero, when the external forces are absent, or driving solutions asymptotically to a pre-assigned compact set in the phase space (attractor). The structure of such attractor depends on the forcing terms p 1 (u, w), p 2 (u, w). It has been observed that thermal dissipation provides substantial damping mechanism for the oscillations so that there may be no need for mechanical dissipation. In fact, such property has been proved for the first time in a special case of a scalar linear plate equation with hinged boundary conditions [26] and in one-dimensional configuration such as thermoelastic rods [23]. However, in the case of free boundary conditions stabilization results in [32,33] do require mechanical dissipation (also for thermal plates) imposed on the boundary of the plate. Only recently it has been shown that in the case of linear thermoelastic plates, uniform decay to zero of the energy can be achieved without any mechanical dissipation, regardless of the boundary conditions [2,3,41,42]. The situation is very different when one considers vectorial structures, including thermoelastic waves. Here no longer one has smoothing property of the dynamics or uniform decay to zero of the energy. The best one can achieve ,without additional mechanical dissipation, is strong stability to zero with a polynomial rate [19,27]. The problem considered in this paper falls into a category of mixed (parabolic-hyperbolic) dynamics with vectorial structure of thermal plates and waves which are nonlinear and strongly coupled. Our aim is to show that nonlinear coupling, while making estimates challenging (due to singularities of nonlinear terms), does provide beneficial mechanism in propagating thermal dissipation onto the entire system -thus forging the desired long-time behavior. The final result is that the dynamics becomes asymptotically finite dimensional and smooth. While this kind of result is to be expected for the dynamics with an overall smoothing effect, it is much less expected in hyperbolic type of models without strong mechanical dissipation and with highly unbounded nonlinear effects. The analysis in this paper illustrates the situation when asymptotic regularity and dissipation can be harvested from thermal effects via boundary traces which become the carriers of the propagation. Since the dynamics of the plate alone is hyperboliclike and unstable, establishing the said propagation is a subtle issue-mainly due to the nature of "free" boundary conditions.
From the above discussion it follows that the combination of free boundary conditions in vectorial structure with the lack of rotational inertia (γ = 0) and strong nonlinearity induced by vectorial structure of the system are the main new features and obstacles in the analysis. The very first result addressing this problem was given in [39] where full von Karman system, without rotational inertia (γ = 0) and with thermal effects was considered. Uniform stabilization to zero with free boundary conditions and boundary dissipation imposed on the in plane velocities was there established. The critical ingredient used for the analysis in [39] is partial thermal smoothing of a single unperturbed trajectory. The present paper takes this analysis to the next level, in the direction of dynamical systems and theory of attractors. This presents new set of challenges mainly due to nonlinear effects which are supercritical (with respect to finite energy space). These prevent the use of known tools in the area of attractors. Nevertheless, we shall show that this strongly nonlinear nonsmooth transient dynamics can be reduced asymptotically to a smooth and finite dimensional set. This will be achieved through a boundary frictional damping applied only to in plane displacements and without any mechanical damping imposed on vertical displacements. The necessity of some mechanical damping imposed on in plane displacements results from well known negative results on the lack of uniform stability in thermal linear waves whenever the dimension of the domain is greater than one [19]. At the technical level, our results critically benefit from the new quasi stability theory [8,13] and "hidden trace regularity" harvested from thermal effects. While [39] provides preliminary road map for the needed estimates, there is a fundamental difference between the theory of attractors dealt with in the present paper and stabilization theory of [39]. While stabilization requires estimates for a single trajectory, theory of attractors require estimates for the differences of trajectories. In the case when nonlinearity of the dynamics is supercritical, single trajectory estimates may still exploit some cancellations. This is not the case with the estimates for the difference of trajectories where the mixing of nonlinearities occurs. Superlinearity does not disappear in the calculations. In order to handle this difficulty, new set of boundary trace estimates will be developed. These estimates are also of independent PDE interest.
1.3. Main results. We begin by introducing some notation. For the norms of standard H s (Sobolev) and L 2 spaces we use: u α,Ω = u H α (Ω) , u α,Γ = u H α (Γ) , and the case α = 0, which corresponds to L 2 spaces, we write u Ω = u L 2 (Ω) and u Γ = u L 2 (Γ) . The corresponding inner products are denoted by (u, v) Ω = (u, v) L 2 (Ω) and u, v Γ = u, v L 2 (Γ) . For α > 0, the space H s 0 (Ω) is the closure of C ∞ 0 (Ω) in H s (Ω), and H −α (Ω) = [H α 0 (Ω)] , where the duality is taken with respect to L 2 (Ω) inner product. Occasionally, by the same symbol , we denote norms and inner products of n-copies of L 2 (O), where O is either Ω or Γ. The same is applied to H α (O). We also consider the following Sobolev spaces The analysis for weak solutions of our system will be done on the (phase) space , and the regularity of solutions will be studied in The assumptions imposed on the external forcing terms p 1 , p 2 are introduced below. Let u = (u 1 , u 2 ) and p 1 (u, w) = p 1,1 (u, w), p 1,2 (u, w) , we assume that there exists a C 2 function P : R 3 → R such that ∇P (u, w) = p 1,1 (u, w), p 1,2 (u, w), p 2 (u, w) , (1.12) and satisfies the following conditions: there exist M, m P 0 such that where M 0 is a positive constant to be defined in (2.25), dependent on σ and on the Korn and Sobolev inequalities. We also assume there exist r 1 and M p > 0 such that, for i = 1, 2, Furthermore, we assume that We note that (1.15)-(1.16) imply that there exists M P > 0 such that The wellposedness and regularity of solutions to our system are given below.
(ii)Regular solutions: Assume that above initial data has further regularity H 1 and appropriate compatibility with respect to the boundary. Then problem (1.1)-(1.10) has a unique regular solution (u, u t , w, w t , φ, θ) ∈ C [0, T ]; In the absence of forcing terms p 1 , p 2 , the wellposedness of problem (1.1)-(1.10) with respect to weak and strong solutions was proved in [39, Theorem 1.1] by using nonlinear semigroup theory along with partial analyticity generated by thermal effects. Since the Nemytskii mapping associated to the forcing p 1 , p 2 , is locally Lipschitz on H, the existence of a unique local solution is granted by semigroup theory (e.g. [10,Theorem 7.2]). Such solutions can be extended to any time interval [0, T ] by using apriori estimate (2.24) below. Remark 1.2. With γ > 0 and absence of thermal effects, Hadamard wellposedness has been proved in [28]. Wellposedness of full von Karman system with thermal effects, γ = 0 and strains accounting for shell's curvature has been recently shown in [46] by resorting to methods of [28].
To establish the existence of attractors one needs the following geometric condition imposed on the "uncontrolled" portion of the boundary Γ 0 . There exists (1.19) Remark 1.3. We note that the geometric assumption in (1.19) is much weaker than the geometric assumption typically imposed in controllability/stabilizability theory [30,32,35]. It involves only uncontrolled part of the boundary Γ 0 , rather than a full boundary ∂Ω.
Our main result reads as follows. Remark 1.4. By assuming additional regularity on the forcing p 1 (u, w), p 2 (u, w) one can reiterate the proof of Theorem 1.2 in order to obtain C ∞ dynamics on the attractor A. See for instance [13,21].
Remark 1.5. The result stated in Theorem 1.2 is also valid with nonlinear damping imposed on u t in (1.4). Instead of u t one can take g(u t ) with g(s) monotone increasing, continuous, g(0) = 0 and subject to: g(s)s M s 2 , |s| > 1 and g(s)s ms 2 for |s| < 1. The above modification will introduce additional technicalities which can be handled as in [12]. In the case when g(s) has unqualified growth at the origin, only the first statement in Theorem 1.2 remains valid.
Remark 1.6. It will be interesting to see whether the result in Theorem 1.2 still holds when (1) Γ 0 has zero measure and (2) in plane displacements do not account for thermal effects φ. We note that in this situation uniform stability to an equilibrium for the unforced plate still holds [39]. However, when studying attractors, items (1) and (2) are needed for the proof of an appropriate unique continuation property. Whether the latter can be proved under weaker assumption, remains an open problem.
The proof of Theorem 1.2 will be given in Section 3. Here, we note, that the key of the proof relies on the following two ingredients: (1) novel abstract criterion in the area of dynamical systems which relies on quasistability property of the dynamical system presented in Section 2 and (2) verification of the abstract condition which depends on new PDE -boundary trace estimates for nonlinear system under consideration. The latter are presented in Section 3.

Energy relations. Along a solution
, t 0, the energy of the system is defined by Here, E k (·) is the kinetic energy defined by and E p (·) is the potential energy given by where the resultant stress N (u, w) is given by N (u, w) = (u) + f (∇w) and It follows that the energy satisfies the identity Indeed, for regular solutions, the proof of (2.21) is standard and follows from classical energy type arguments. For weak solutions the energy function satisfies the inequality. However, due to the uniqueness of weak solutions, one also shows by convexity methods [39] that actually (2.21) holds for all weak solutions. Next we establish a relation between E y (·) and E y (·). To this end, we note that for u ∈ H 1 (Ω), the Korn inequality together with Sobolev embedding give The following lower bound is critical.
Remark 2.1. We note that the potential energy E p (·) is topologically equivalent 2 and therefore E y (·) is topologically equivalent to the space H.

Tensor identities.
In order to simplify the verification of some rather long calculations, we provide a few elementary tensor identities. Let us define the vector field: Then we have that where R is a tensor given by Given two (fourth order) tensors A, B, written as 4-vectors, we define ( and symmetric coefficients c j,i . Then (2.28) Now, taking A = σ[ (u)] and using identities (2.26) and (2.27), we obtain 3. Dynamics of quasi-stable systems. In this subsection we provide recent results pertaining to long time behavior of quasi-stable systems [8,13]. These results are critical for the development, since classical approaches in dynamical system theory based on decomposition of trajectories [22,31,45,51] are not applicable within the context of supercritical nonlinearities. We begin with the following classical result, cf. [4,12,13,22,31,51].
Theorem 2.1. Let (H, S(t)) be a dynamical system, dissipative and asymptotically smooth. Then it possesses a unique compact global attractor A ⊂ H.
Another type of dissipativeness is characterized by gradient systems, that is, systems possessing a strict Lyapunov function. In other words, there is a functional Φ : Regarding the structure of the attractors we know that M + (N ) ⊂ A, where N is the set of stationary points of {S(t)} and M + (N ) is the unstable manifold of y ∈ H such that there exists a full trajectory u(t) satisfying For gradient systems it is possible to prove that the unstable manifold M + (N ) coincides with the attractor A. The following result is well-known. See for instance [13,Corollary 7.5.7].
Theorem 2.2. Let (H, S(t)) be an asymptotically smooth gradient system with the corresponding Lyapunov functional denoted by Φ. Suppose that and that the set of stationary points N is bounded. Then (H, S(t)) has a compact global attractor which coincides with the unstable manifold M + (N ).
Remark 2.2. The advantage of the theorem above is that for gradient systems an existence of global attractor does not require proving existence of an absorbing ball -a task that can be technical and cumbersome.
Our aim is to establish existence of a global attractor along with the properties such a finite-dimensionality and smoothness. In order to achieve this we shall exploit the concept of quasistability -Definition 7.9.2 in [13, Chapter 7] -which allows to prove such properties in "one shot" provided one has "good" estimates for the differences of two trajectories originating in a bounded set B ⊂ H.
Let X, Y, Z be three reflexive Banach spaces with X compactly embedded into Y , and define H = X × Y × Z. Suppose that (H, S(t)) is a dynamical system of the form where the functions u and ξ have regularity Then we say that (H, S(t)) is quasi-stable on a set B ⊂ H, if there exists a compact semi-norm n X on X and nonnegative scalar functions a(t) and c(t), locally bounded for any y 1 , y 2 ∈ B. In this case the following result holds.
Remark 2.3. Quadratic dependence of the compact term in the inequality (2.34) is critical. In fact, achieving this quadratic dependence is one of the main technical difficulties of the problem. We note that in order to obtain just an existence of compact attractor, a much weaker form of this inequality suffices. In particular, there is no restriction on the power of compact term (could be sublinear). The most useful property of quasi-stable systems is that quasistability on the attractor implies automatically smoothness and finite-dimensionality of the said attractor. This fact is stated in theorem below.
where M depends on c(t).
By summarizing the results stated in Theorem 2.2 which guarantees the existence of a global attractor, and Theorem 2.4, which provides finite fractal dimension and smoothness of the said attractor, we arrive at: Corollary 2.1. Let (H, S(t)) given by (2.31) and satisfying (2.32) be a quasistable, gradient system with Lyapunov function satisfying (2.30) and a bounded set of stationary points. Then (H, S(t)) admits a finite-dimensional global attractor A which is also "smooth": 3. Global attractors-proof of Theorem 1.2. This section is devoted to the proof of the main result formulated in Theorem 1.2. This is based on the application of Corollary 2.1. To this end we must show that (H, S(t)) is: (1) gradient system with a Lyapunov function satisfying (2.30), and (2) quasi-stable system with the appropriate bounds for c(t). The property of gradient system relies on a new unique continuation property shown for the system under consideration. The property of quasi-stability is the most technical part of the proof which requires deep PDE results related to hidden regularity of the boundary traces corresponding to vectorial systems with free boundary conditions. These results are of independent PDE interest.
3.1. Proving quasi-stability. In this subsection we shall prove that our problem is quasi-stable. Accordingly, we must show that the difference of two trajectories satisfies estimate (2.34). To this end one needs rather extensive background and several energy estimates. This will be established in five subsections.
3.1.1. Comparing two trajectories. Let B be a bounded set of H and consider two solutions of (1.1)-(1.10), solves the problem, 3) where with the corresponding initial datã The unperturbed energy of (3.3)-(3.7) is defined bỹ Then we have the following energy equality, where Remark 3.1. We verify condition (2.34) by obtaining the following estimatẽ for suitable constants C, β, ε > 0. This will be achieved in Lemma 3.7.
We end this subsection with some estimates for f (∇w i ), i = 1, 2.

A first observability inequality.
Here we obtain an observability inequality that reconstructs the integral of the linear energy in terms of the dissipation, lower order terms and also boundary traces, which are not apriori bounded by the energy. The estimate will be obtained by multipliers method applied to all three components of the system [2,39]. In order to control these boundary terms, more subtle estimates will be needed which invoke partially regularizing effect of thermoelasticity as well as micro local estimates applied to a hyperbolic component represented by u. This will be done in Subsection 3.1.3.
Proof. The proof of this lemma is divided into several steps. The geometric condition (1.19) will be used.
Step 1. Estimate for kinetic energy of in-plane displacement: Multiplying both sides of equation (3.3) by h∇ũ, where h(x) = x − x 0 , and integrate in time and space. We find Integrating by parts in time and using divergence theorem yield Applying the divergence and Gauss theorems in the second term of (3.9), we obtain Note that Then, the identity together with boundary condition (3.6) 2 imply that It follows from identity (2.29) that which combined with (2.28) and with Gauss theorem implies that (3.13) Consequently from (3.11)-(3.13) we find that (3.14) Combining (3.10) and (3.14) with (3.9) we obtain Let us estimate the nonlinear term P 1 (ũ,w). Using the assumption (1.15) we find that Then Hölder's inequality with (r−1) 2(r+1) + 1 r+1 + 1 2 = 1 implies that Here we used the fact that T 0 σ[ (ũ)], (ũ)h·ν Γ1 dt C Σ1 |∇ũ| 2 dΣ 1 .
Step 2. Estimate for the difference of potential and kinetic energies: Multiply both sides of equation (3.3) byũ and integrate in time and space Using Gauss theorem in the second term of (3.18) we find Boundary conditions (3.6) 1 and (3.6) 2 imply that These identities in (3.18) imply in the following equality Proceeding as in (3.16) we obtain the following estimate Choosing δ > 0 small enough and using Lemma 3.1 we find  To handle the second term in (3.20) we use the following identity where ψ ∈ H 2 (Ω). Taking ψ =w and using boundary conditions (3.6) 1 , (3.6) 3 ,(3.6) 4 we find Using Gauss theorem we can rewrite the third term of (3.20) as

(3.24)
Let us estimate R. Using the definition of stress N (·, ·) we find The inequality u ⊗ v Ω C u ε,Ω v 1−ε,Ω , which holds for ε ∈ (0, 1), implies that Step 3. Estimate for kinetic energy of vertical displacement: Let us consider the following operators.
Proceeding as before, we obtain Using relation (3.27) we obtain On the other hand, for every δ, δ 0 > 0, there exist constants C δ , C δ0 > 0 such that It remains to estimate the nonlinear terms in (3.28). For this, considering the definition of N 2 , we find Let us estimate the integrals on the right-side of (3.35). Proceeding as in (3.25) we obtain Then we have (3.37) Step 4. Completion of the proof: Combining the inequalities (3.17), (3.26), (3.37) and selecting suitable δ > 0 small we obtain the conclusion.

3.1.3.
Trace regularity and analytic estimates. In order to control boundary terms in estimate given in Lemma 3.2, more subtle estimates are needed, including trace regularity and analytic estimates. They are essential to prove the quasistability inequality. Our result is based on the corresponding trace estimate for the linear model of dynamic elasticity [24]. The analytic estimates rely on the analyticity of the semigroup generator associated with the linear thermoelastic plate. Then for any ε ∈ (0, 1 4 ) and α ∈ (0, T 2 ) the following trace regularity is valid.
Proof. The proof is divided into several steps.
Step 5. Conclusion: Integrating in time and space the inequality (3.45) we obtain

Inequalities (3.39), (3.40) and (3.44) imply
Σα Then, inequality (ii) of Lemma 3.1 implies in the assertion of Lemma 3.3. Next we prove an improved regularity for the vertical displacementw. This is done by exploiting the analyticity of the thermoelastic semigroup. Proof. The proof of the lemma is divided into three parts.
Step 1. Abstract setting: We rewrite the original problem via variation of parameters. To accomplish this we introduce the following operators.
• The biharmonic operator: Let A M be a positive and self-adjoint operator on L 2 (Ω) given by • The Green's operators: Let G i , i = 1, 2, be the operators corresponding to the mechanical boundary conditions defined by and . Elliptic regularity (e.g. [43]) gives G 1 : L 2 (Γ 1 ) → H We have that Then from the definition of operators A M , G 1 , G 2 and A, we have, (cf. [41]) Therefore, we can rewrite the problem for (w,θ) in the following form where A : The operator A generates an analytic and exponentially stable semigroup on the space cf. [41]. Moreover A is m-dissipative and A −1 is bounded in H. Therefore, from [6, Proposition 6.1] we infer that, for α ∈ (0, 1), (3.48) Then, for ε < 1 2 , we can rewrite the solution of (3.47) using variation of parameters formula, (3.49) where F(ũ,w,φ) = div{F (ũ,w,φ)} + A M G 2 (F (ũ,w,φ)·ν) − P 2 (ũ,w).
Lemma 3.6. One has  Using estimates of Lemma 3.1 we find that (3.64) We estimate the three integrals in (3.64). Integrating by parts in time we obtain ∇w Ω dtds. Integration by parts in space variable and the Trace Theorem imply that Hölder inequality and Sobolev embedding imply that To conclude, we have to estimate the third integral in (3.64). Integration by parts in time and space, and the fact that |∇w| 2 1,Ω = ∇w·∇w 1,Ω C w 1,Ω w 2+ε,Ω , we obtain Proof. Inserting estimate from Lemma 3.6 into (3.56) and using interpolation inequality we find that Now let δ > 0 be small enough. For T > 4C δ = T 0 and α = C δ < T 2 we have that (3.73) Next, we estimate the damping term D T 0 (ũ,φ,θ). Energy equality (3.8) and estimate from Lemma 3.6 imply that Proof. By using an isomorphism, we can reorder the components of a trajectory as (u, w, u t , w t , φ, θ). That is, we can assume S(t) : Then conditions (2.31), (2.32) and (2.33) are clearly satisfied. Let show that (2.34) also holds. To this end, we consider a X-seminorm defined by, where b(t) = C 1 e −βt and c(t) = C 2 . This proves that our system is quasi-stable on B with the c(t) independent on time t > 0 -as desired.
3.2. Gradient systems and completion of the proof of the Theorem 1.2.
The proof of Theorem 1.2 will follow from Theorem 2.2. To accomplish this we need to establish gradient structure of system (H, S(t)). We shall take the energy functional E y as a Lyapunov function Φ(y), where E y corresponds to the energy at the point y defined by (2.20). Thus Φ(S(t)y) = E S(t)y of the trajectory S(t)y with a given initial data y ∈ H. From (2.21) it follows that t → Φ(S(t)y) is decreasing for any y ∈ H. The fact that Lyapunov function is strict follows from the following Unique Continuation Property formulated in the Lemma below.
(3.89) Thus (3.87) is overdetermined on the boundary wave equation with a potential satisfying regularity assumption in (3.89). We appeal now to [47] [ or Theorem 1.2 in [25] and [20] ] to claim thatū = u t ≡ 0 in Ω. Thus, the dynamics has been reduced to a stationary elliptic problem.
Remark 3.3. It is interesting to note the role played by the second thermal variable φ. In the previous calculations [leading to quasistability] the additional dissipation due to φ did not play any major role. However, when dealing with weak solutions which are overdetermined on the boundary, the condition div{u t } = 0 -resulting from the dissipation in φ variable, allows to reduce system of dynamic elasticity to a classical wave equation (3.87) with Cauchy zero data on Γ 1 . For such equation UCP property has been shown [47] for just L 2 solutions (as in our case). Otherwise, one would need to introduce appropriate approximations of overdetermined problems (as in Proposition 2.1 in [38], or [25]) which would allow to deduce additional regularity of the overdetermined problem. This, however, will make the analysis much more technical -see [37], Section 6). Multiplying (3.90), (3.91) by u, w, respectively, and integrating over Ω, we obtain 1 2 Ω σ[N (u, w)]N (u, w)dΩ + 1 2 a(w, w)+ κ 2 Γ1 |u| 2 dΓ 1 = −