ASYMPTOTIC BEHAVIOR OF A HYPERBOLIC SYSTEM ARISING IN FERROELECTRICITY

. We consider a coupled hyperbolic system which describes the evolution of the electromagnetic ﬁeld inside a ferroelectric cylindrical material in the framework of the Greenberg-MacCamy-Coﬀman model. In this paper we analyze the asymptotic behavior of the solutions from the viewpoint of inﬁnite- dimensional dissipative dynamical systems. We ﬁrst prove the existence of an absorbing set and of a compact global attractor in the energy phase-space. A suﬃcient condition for the decay of the solutions is also obtained. The main diﬃculty arises in connection with the study of the regularity property of the attractor. Indeed, the physically reasonable boundary conditions prevent the use of a technique based on multiplication by fractional operators and boot- strap arguments. We obtain the desired regularity through a decomposition technique introduced by Pata and Zelik for the damped semilinear wave equa- tion. Finally we provide the existence of an exponential attractor.


1.
Introduction. In this work we investigate the asymptotic behavior of a coupled hyperbolic system of the form u tt + u t + p t − ∆u = 0 p tt + p t − u t − ∆p + p + φ(p) = 0 (1.1) in Ω × (0, ∞), where Ω ⊂ R 2 is a bounded and connected domain with smooth boundary ∂Ω. This system of equations governs the evolution of the electromagnetic field inside a ferroelectric material occupying the cylindrical region Ω× R, according with a physical model recently proposed by Greenberg et al. (see [10]). In this model u represents a field connected to the electric field ε (directed along the z axis) and to the components of the magnetic field h 1 , h 2 (lying in the x, y plane) by the relations c being the velocity of light in vacuum, while p is the polarization field inside the material (directed along the z axis). In [10] the nonlinearity φ is a smooth globally lipschitz function satisfying a certain coercivity property. For the sake of more generality, here we shall assume φ with polynomial growth of finite and arbitrary order and all the results will be deduced under this general assumption. In [10] a quite detailed analysis of the stationary states is performed. However, the study of the asymptotic behavior is still at a preliminary stage. In the present paper we want to deepen the analysis of the long time behavior by studying global asymptotic properties such that the existence of a bounded absorbing set (thus showing the dissipative feature of the system), of the global attractor and its regularity, as well as the existence of an exponential attractor of optimal regularity. The results will be obtained under the physically significant Dirichlet-Neumann boundary conditions u = ∂ n p = 0, on ∂Ω × (0, ∞), ∂ n being the outward normal derivative. The Dirichlet-Dirichlet boundary conditions u = p = 0, on ∂Ω × (0, ∞) can also be considered. Indeed, these latter conditions, although less significant from the physical point of view, allow to obtain more easily some results concerning the regularity of the attractor and the existence of the exponential attractor, through the use of standard techniques. Such results will be obtained in this paper under the Dirichlet-Neumann boundary conditions exploiting a recent approach due to Pata and Zelik (see [16]) based on suitable decomposition of the evolution semigroup.
The formal problem we want to analyze is therefore the following Problem P. Find (u, p) solution to the system (1.1) in Ω × (0, ∞) with boundary conditions u = ∂ n p = 0 on ∂Ω×(0, ∞), and initial conditions u(0) = u 0 , u t (0) = u 1 , p(0) = p 0 , p t (0) = p 1 in Ω. This paper is organized in the following way. In Section 2 we introduce some functional notation and state Gronwall type lemmas useful in the following. Section 3 is devoted to the weak formulation of Problem P and to the well-posedness result. In particular, we associate with P a strongly continuous semigroup acting on the energy phase-space H 0 . In other words, P is interpreted, in the theory of infinite dimensional dynamical systems, as a generator of trajectories in the energy phasespace. This result, as well as the nonlinear feature of the system (causing the great sensibility to the initial conditions) suggests that the correct approach to study its asymptotic dynamics is geometric. In Section 4 the dissipativity of the system is deduced by proving the existence of a bounded absorbing set on the energy phase-space, as well as the existence of a bounded absorbing set in the second order phase-space H 1 . In this section we also deduce a sufficient condition on the nonlinearity φ which ensures the uniform decay of the trajectories departing from every bounded subset of H 0 (see Proposition 4.1). The existence of the global attractor is shown in Section 5. The remarkable fact is that this result can be obtained by means of the same hypotheses used for the well-posedness result in H 0 (see (H1)-(H3)). No further assumption is needed. In Section 6 we consider the regularity of the attractor, for the Dirichlet-Neumann case, obtained, adding further regularity assumptions on the nonlinearity, through the use of the Pata-Zelik technique. Such technique can obviously be applied for the Dirichlet-Dirichlet case as well. The last section is devoted to the existence of an exponential attractor. For this purpose we exploit the abstract result due to Efendiev, Miranville and Zelik (see [6,Proposition 1]) concerning the existence of exponential attractors for evolution equations.
2. Functional setting and notation. We denote by (·, ·) and · the inner product and the norm on L 2 (Ω), respectively. The symbol ·, · will stand for the duality pairing between a Banach space and its dual. We introduce the operators A = −∆ + I and B = −∆ (∆ is the laplacian in two dimensions) with domains D(A) = {v ∈ H 2 (Ω) : ∂ n v = 0 on ∂Ω} and D(B) = H 2 (Ω) ∩ H 1 0 (Ω), respectively. As is well known, A and B are (unbounded) linear, strictly positive, self-adjoint operators with compact inverse. The spectral theory of this class of operators allows defining, for all s ∈ R, the fractional operators A s and B s , with domains D(A s ) and D(B s ). Identifying L 2 (Ω) with its dual L 2 (Ω) ′ , for any s ∈ R we consider the two families of Hilbert spaces V s := D(A s/2 ) and V s 0 := D(B s/2 ) with the natural inner products and norms. We recall that V s = V s 0 for s ∈ (−1/2, 1/2). Moreover, 0 , for s ≥ 0 and the compact and dense injections V s ֒→֒→ V r , V s 0 ֒→֒→ V r 0 , for s > r. In particular, denoting by λ A and λ B the first eigenvalues of A and B respectively, we have the inequalities Formula (2.2), for r = 0 and s = 1, is the Poincaré inequality. Concerning the phase-space for our problem, we introduce, for s ∈ R, the product Hilbert spaces , with corresponding norms, induced by their inner products, given by z 2 s := z 2 Hs = B (1+s)/2 u 2 + B s/2 v 2 + A (1+s)/2 p 2 + A s/2 q 2 for all z = (u, v, p, q) ∈ H s . In particular we have . From Section 5 on, we denote by c ≥ 0 a generic constant, that may vary even from line to line within the same equation, depending on Ω and φ. Further dependences will be specified when necessary. Furthermore, we will use, sometimes without explicit reference, relations (2.1) and (2.2) as well as the Young and generalized Hölder inequalities and the usual Sobolev embeddings. We conclude this section with two technical lemmas that will be needed in the course of the investigation. Lemma 2.1. Let X be a Banach space, and Z ⊂ C([0, +∞); X). Let be given a functional E : X → R such that sup t≥0 E(z(t)) ≥ −m and E(z(0)) ≤ M , for some m, M ≥ 0 and for every z ∈ Z. In addition assume that E(z(·)) ∈ C 1 ([0, +∞)) for every z ∈ Z and that the differential inequality holds for all t ≥ 0 and for some δ 0 > 0, k ≥ 0, both independent of z ∈ Z. Then, for every ǫ > 0 there is t 0 = t 0 (M, ǫ) ≥ 0 such that, for every z ∈ Z Furthermore, the time t 0 can be expressed by t 0 = (M + m)/ǫ.
For the proof, we refer the reader to [3, Lemma 2.1].
3. Well-Posedness. The assumptions we make on the nonlinearity are the follow- is a weak solution to P in I with initial data z 0 provided that for every v 0 ∈ V 1 0 and v ∈ V 1 , almost everywhere in I, and u(0) = u 0 , u t (0) = u 1 , p(0) = p 0 , p t (0) = p 1 , almost everywhere in Ω.
Existence, uniqueness and continuous dependence, according to Definition 3.1, can be deduced by standard arguments based on a Faedo-Galerkin approximation scheme and on the use of the energy identity. We shall obtain energy type estimates in Section 4 (see (4.1), (4.14)). More precisely, we have the following well-posedness result Theorem 3.2. In the hypotheses (H1)-(H3), for any T > 0, Problem P has a unique (weak) solution z on the time interval I = [0, T ] with initial data z 0 ∈ H 0 . Moreover, if z 01 and z 02 are two sets of data in H 0 , and z 1 and z 2 are the two corresponding solutions on [0, ∞), there exists θ 0 > 0, depending (continuously and increasingly) only on the H 0 -norms of the data z 0i 0 for i = 1, 2 (besides on Ω and φ) such that Since the system is autonomous, the well-posedness result immediately leads to the following Corollary 3.1. The one-parameter family of continuous (nonlinear) operators S(t) : H 0 → H 0 defined by S(t)z 0 := z(t), for every t ≥ 0 and every z 0 ∈ H 0 , where z(t) is the solution to P at time t with z(0) = z 0 , is a strongly continuous semigroup on the phase-space H 0 .
Remark 3.2. The particular time dependence in the estimate (3.3) holds if either (H3) or (H3 * ) with the further assumption α < λ A /2 are fulfilled. Indeed, in this case from the energy identity (see (4.14)) we easily get the control z(t) 0 ≤ Λ( z 0 0 ) for every t ≥ 0 (see also Corollary 4.1). The assumption (H3 * ) with α ≥ λ A /2 still yields well-posedness, but the time dependence in the estimate (3.3) is more involved.
Furthermore, due to standard regularity results, it is possible to prove that, under the assumptions (H3) and is also a strongly continuous semigroup on the phase-space H 1 . In particular, the continuous dependence estimate holds in H 1 in the following form: for every R 1 , T > 0 there exists for every t ∈ [0, T ] and every z 1 , z 2 ∈ H 1 with z 1 1 , z 2 1 ≤ R 1 .

Dissipativity.
Here we would like to see whether the trajectories originating from any given bounded subset of the phase-space H 0 eventually fall, uniformly in time, into a fixed bounded subset, which we call absorbing set. For this purpose we shall need some uniform (with respect to time) energy type estimates.
Theorem 4.1. Let (H1)-(H3) hold. Then, there exists a constant R 0 > 0 with the following property: given any R > 0, there exists t 0 = t 0 (R) such that, whenever z 0 0 ≤ R, the inequality S(t)z 0 0 ≤ R 0 holds for every t ≥ t 0 (R). Consequently, the set is a bounded absorbing set for the semigroup S(t) generated by Problem P on H 0 , that is, for every given bounded subset B ⊂ H 0 , there exists a time t Proof. We suppose to work within a proper approximation scheme, with regular data and solutions, to justify the formal estimates we derive below. We multiply (1.1) 1 and (1.1) 2 in L 2 (Ω) by the auxiliary variables ξ := u t + ǫu and ζ := p t + ǫp, respectively, with 0 < ǫ ≤ ǫ 0 and ǫ 0 to be chosen later. Adding together the resulting equations, we get 1 2 By (H3) it is easy to show that there exists ν ∈ (0, 1] such that We now introduce the following functional E : for every z = (u, v, p, q) ∈ H 0 . Using (4.3) and the Young inequality, it is not difficult to show that provided that ǫ 0 is small enough, say ǫ 0 ∈ (0, ǫ ′ 0 ] (see Remark 4.1 below). From (H2), by means of relations (2.1), (2.2) and by the Sobolev embedding H 1 (Ω) ֒→ L r (Ω) for every r ∈ [1, +∞) (N = 2), we also get By virtue of (4.2) and taking into account the definition of the functional E, from (4.1) we obtain 1 2 Using again the Young inequality and choosing ǫ 0 small enough, namely ǫ 0 ∈ (0, ǫ ′′ 0 ] (cf. Remark 4.1 again), we obtain from (4.7) 1 2 We now have provided we choose ǫ ∈ (0, ǫ ′ 0 ] small enough. Therefore, by means of (4.9), inequality where δ 0 = ǫ 0 ν/3 and ǫ 0 = min{ǫ ′ 0 , ǫ ′′ 0 }. The existence of a bounded absorbing set is now a direct consequence of (4.10), in light of Lemma 2.1. Actually, let us fix R > 0 and a set of initial data z 0 ∈ H 0 , with z 0 0 ≤ R. By (4.6) we have the bound (4.11) We now take X = H 0 and Z ⊂ C 0 ([0, +∞); H 0 ) given by the family of the trajectories departing from the initial data z 0 ∈ H 0 with z 0 0 ≤ R. From Lemma 2.1 we therefore conclude that there exists a time t 0 = t 0 (R) > 0 such that for every t ≥ t 0 and for every z 0 ∈ H 0 with z 0 0 ≤ R. Here we have set z(t; z 0 ) := S(t)z 0 . Estimating the right hand side of (4.12) by means of (4.6) and taking account of (4.5), we conclude that there exists R 0 > 0 such that Another important corollary, that will be useful in the following, provides the uniform control of the dissipation integral, namely, Proof. We write (4.1) for ǫ = 0 getting the energy identity Integrating with respect to time and using (4.3) we are led to From this inequality, letting t → ∞, we deduce the thesis.
Remark 4.1. We can furnish an estimate for the radius R 0 of the absorbing set B 0 in terms of the parameters of the problem. First let us introduce the following notation. Given a 1 , a 2 , a 3 ≥ 0 and r ≥ 2, we denote by γ = γ(a 1 , a 2 , a 3 ; r) a nonnegative constant such that a 1 t r + a 2 t 2 + a 3 t ≤ γ(t + t r ) for every t ≥ 0. We indicate by c 0 a nonnegative constant such that |F (s)| ≤ c 0 (|s|+|s| r ) for every s ∈ R.
With the previous notation we can also take c 0 = γ(c 0 /2, c 0 /2, |φ(0)|; r), where c 0 is the constant appearing in (H2). For the constants ν, c 1 and c 2 in (4.2), (4.3) we can is the smallest integer greater or equal to x, for every x > 0, and P : is a bounded and linear extension operator). The constant c i depends on Ω and r. Finally, the values of ǫ ′ 0 , ǫ ′′ 0 can be given by We observe that in (4.6) the constant c 3 is given as above, provided ǫ ≤ ǫ ′ 0 . From (4.5), (4.6), (4.12) we thus can deduce the required estimate Remark 4.2 (Uniform decay of the trajectories). Exploiting the considerations of the previous remark and using Lemma 2.1 once more, we can immediately deduce a sufficient condition which ensures the decay in H 0 of the trajectories uniformly from every bounded subset B ⊂ H 0 . Indeed, let η > 0 be fixed arbitrary. Then, the quantity R 0,η > 0 given by is the radius of an absorbing ball B 0 = B H0 (0, R 0,η ). Therefore, for every R > 0, there exists a time t 0 = t 0 (R, η) > 0, such that, for every t ≥ t 0 (R, η) we have z(t; z 0 ) 0 ≤ R 0,η for any z 0 ∈ H 0 with z 0 0 ≤ R. From Lemma 2.1, (4.5) and (4.6) we can also infer t 0 = [c 3 R(1 + R r−1 ) + c 2 ]/η. We hence recognize at once the desired sufficient condition, that is c 1 = c 2 = 0, and we can state the following proposition  We conclude this section with the result concerning the existence of a bounded absorbing set in H 1 for the semigroup S(t) : H 1 → H 1 . This will be guaranteed by the next theorem whose proof, based on the generalized Gronwall lemma recalled in Section 2 (see Lemma 2.2), requires, besides (H3)-(H5), the following further (but reasonable) assumption (H6) φ ′ (s) ≥ −l, ∀s ∈ R, l ≥ 0.
Theorem 4.2. Let (H3)-(H6) hold. Given R 0 ≥ 0 and R 1 ≥ 0 such that z 0 0 ≤ R 0 and z 0 1 ≤ R 1 , there exist constants C = C(R 0 ), K = K(R 1 ), depending increasingly and continuously on R 0 and R 1 respectively, and ǫ 1 > 0 such that, for every t ≥ 0, Proof. We rewrite system (1.1) in the form where ψ(s) := φ(s) + ls. For z 0 ∈ H 1 , we consider the linear nonhomogeneous problem obtained by differentiation of the above system with respect to time. Since we have (v 0 , v 1 , q 0 , q 1 ) ∈ H 0 , by virtue of standard well-posedness results for linear equations (see, e.g., [17]), problem (4.17) admits a unique weak solution w := (v, v t , q, q t ) such that w ∈ C 0 ([0, ∞); H 0 ). By comparision with the solution to Problem P we obtain v(t) = u t (t), q(t) = p t (t) and hence w(t) = z t (t). Now, for ǫ > 0 to be determined later, we multiply in L 2 (Ω) (4.17) 1 by ξ := v t + ǫv, (4.17) 2 by ζ := q t + ǫq and we add together the resulting equations. After some calculations we obtain If we define it is easy to see that there holds provided ǫ is small enough, where k 1 and k 2 are positive constants, with only k 2 depending on the H 0 -norm of the initial data z 0 . To prove (4.20) one uses (H2), the control z(t) 0 ≤ Λ( z 0 0 ) (provided by Corollary 4.1), Young inequality and the fact that ψ ′ (s) ≥ 0 for every s ∈ R. From now on, in the course of this proof, we denote by k some positive constant depending, increasingly and continuously, on z 0 0 . After exploiting Corollary 4.1 to estimate the last three terms on the right hand side of (4.18) as we can write for ǫ small enough Finally, using (H5) and Corollary 4.1 once more, the first term on the right hand side of (4.18) can be estimated as follows  with c(·) a nondecreasing continuous function, whereas using (4.16) 1 and (4.16) 2 and the fact that z 0 0 ≤ R 0 , for every t ≥ 0, we have The thesis follows from (4.25), (4.26) and (4.27).

5.
The global attractor. The aim of this section is to prove the existence of the global attractor for the semigroup S(t) on H 0 . We consider Problem P, that is system (1.1) endowed with Dirichlet-Neumann boundary conditions. Nevertheless the same procedure, with few modifications, can be repeated in the case of Dirichlet-Dirichlet boundary conditions as well. We recall that the global attractor is the (unique) compact set A ⊂ H 0 which is fully invariant, i.e., S(t)A = A for every t ≥ 0, and attracting in the sense of the Hausdorff semidistance. See, for instance, [1,11,12,17] for reference on the general theory of dissipative infinite-dimensional dynamical systems. Let us state the main result of this section. Proof. We decompose the solution z = (u, u t , p, p t ) to P with initial data z 0 = (u 0 , u 1 , p 0 , p 1 ) ∈ H 0 as z = z d + z c , where z d = (u d , ∂ t u d , p d , ∂ t p d ) and z c = (u c , ∂ t u c , p c , ∂ t p c ) are the solutions to the problems   It is easy to check that problems (5.1) and (5.2) are well posed. The thesis of the theorem follows from the general theory of dynamical systems once we show that, uniformly for z 0 ∈ B 0 z d (t) → 0 in H 0 as t → +∞, whereas (for z 0 ∈ B 0 ) z c (t) lies in a compact subset of H 0 (possibly depending on t) for all t ≥ 0. This is precisely the content of Lemma 5.2 and Lemma 5.3. Proof. We multiply (5.2) 1 by B s ∂ t u c and (5.2) 2 by A s ∂ t p c , where we fix s ∈ (0, 1/2). Adding the resulting identities we are led to the differential equality ).

(5.3)
Observe now that where we have exploited the embedding D(B s/2 ) ֒→ D(A s/2 ), for every s > 0, from which follows D(A −s/2 ) ֒→ D(B −s/2 ), for every s > 0. Moreover where we have used the fact that, for s ∈ (0, 1/2), D(A s/2 ) = D(B s/2 ) and the operator A −s/2 B s/2 is bounded (i.e., A −s/2 B s/2 w ≤ c w for every w ∈ D(B s/2 )). Integrating (5.3) with respect to time from 0 to t and taking into account (5.2) 3 , (5.4), (5.5) we get Now, we integrate by parts the last term on the right side of (5.6) and get It is easy to see that the first term on the right side of (5.7) is bounded. Indeed, for z 0 ∈ B 0 , denoting henceforth by c a nonnegative constant depending on R 0 (besides on Ω and φ), we have A s p c (t) ≤ c A 1/2 p c (t) ≤ c, for all t ≥ 0, as a consequence of the decay to zero of p d (Lemma 5.2), and of Corollary 4.1 (which implies where we have used the embedding D(A (1−s)/2 ) ֒→ L 2/s (Ω) and the fact that, for z 0 ∈ B 0 , φ ′ (p) L 2/(1−s) , p t ≤ c, by virtue of Corollary 4.1. Substituting (5.8) into (5.6), we obtain the differential inequality The standard Gronwall lemma yields Φ(t) ≤ c(t) for all t ≥ 0, where c(t) generally depends on t, but it is independent of z 0 (provided z 0 ∈ B 0 ). Hence z c (t) ∈ B Hs (0, c(t)), for every t ≥ 0 and every z 0 ∈ B 0 . Hence, the thesis follows from the compact embedding H s ֒→֒→ H 0 .

6.
Smooth attracting sets. We now establish the existence of a bounded subset in H 1 , denoted by B 1 , which attracts the bounded subsets in H 0 exponentially fast. This circumstance, on one hand, will provide the regularity of the attractor, and, on the other hand, will turn to be very useful in the construction of the exponential attractor. We point out that, through the use of a different proof of Lemma 5.3 obtained by multiplying (5.2) 1 and (5.2) 2 by B s ∂ t u c + ǫB s u c and A s ∂ t p c + ǫA s p c , respectively, and taking ǫ small enough, it can be shown that the solution z c of (5.2) actually fulfills This implies that the global attractor A is a bounded subset of the Hilbert space H s , for s ∈ (0, 1/2), which is compactly embedded into the phase-space H 0 . In this section we show that the bounded inclusion A ⊂ H s can be pushed up to s = 1. For this purpose, the application of the technique based on the multiplication by fractional operators and on bootstrap arguments, which works perfectly for the the system (1.1) with Dirichlet-Dirichlet boundary conditions, is problematic in the case of the Dirichlet-Neumann boundary conditions, due essentially to the different domains of A s/2 and B s/2 for s ≥ 1/2, and to the presence of the coupling terms. In order to achieve the existence of B 1 and hence the H 1 -regularity of the attractor for Problem P we therefore employ a different approach, which consists in the application to system (1.1) of a new technique due to Pata and Zelik (see [16]), whose key step is a suitable decomposition of the solution semigroup. This decomposition has been recently employed successfully in other recent works (see, e.g., [8,9,18]). Notice that this procedure can also be intended as a proof of existence of the global attractor A. Nevertheless, the only existence of A can be deduced under weaker assumptions, like those of Section 5 and Section 4. Following [16], we shall need, besides (H3), the assumptions (H4), (H5) and (H6). Here is the result we want to prove Theorem 6.1. Let (H3)-(H6) hold. Then, there exists a subset B 1 ⊂ H 1 closed and bounded in H 1 and ν > 0 such that for some M > 0 depending on R 0 .
As a straightforward consequence, by virtue of the minimality property of the global attractor, we have the following Corollary 6.1. In the hypotheses (H3)-(H6), the global attractor A of the semigroup on H 0 associated with Problem P is contained and bounded in H 1 .
In order to prove Theorem 6.1 we consider the initial data z 0 ∈ B 0 and we decompose the solution z to P into the sum z = z 1 + z 2 , where z 1 = (u 1 , u 1t , p 1 , p 1t ) and z 2 = (u 2 , u 2t , p 2 , p 2t ) are the solutions to the problems, respectively, Here we have set ψ(s) := φ(s) + θs with θ ≥ l, in order to have, by (H6), ψ ′ (s) ≥ 0. The following lemmas will be needed for the proof of Theorem 6.1. We stress that c ≥ 0 stands for a generic constant depending possibly only on R 0 (the radius of the absorbing set) and on Ω and φ, but neither on z 0 ∈ B 0 nor on the time t. Lemma 6.2. We have z 2 (t) 0 ≤ c, for every t ≥ 0.
Proof. We can use the same argument of the proof of Theorem 4.1. Indeed, identity (4.1) can be obviously rewritten for system (6.3), replacing F (p) with Ψ(p 2 ) := p2 0 ψ(σ)dσ on the left hand side of (4.1) and adding the additional term θ(p, ζ 2 ) on the right hand side. It is immediate to verify that (H3) is still fulfilled with φ replaced by ψ and hence (4.3) still holds for Ψ(p 2 ) in place of F (p). Writing θ(p, ζ 2 ) ≤ 1 2 ζ 2 2 + c p 2 , we are led to d dt E(z 2 (t)) + δ 0 z 2 (t) 2 0 ≤ 2ǫ 0 c 1 + c p 2 ≤ c for z 0 ∈ B 0 . By means of Lemma 2.1 (observe that here z 2 (0) = 0) we immediately conclude the proof. Lemma 6.3. For every 0 ≤ s ≤ t and every ω > 0 we have Proof. We multiply (6.3) 1 by u 2t and (6.3) 2 by p 2t . Adding the resulting equations we get d dt it is easy to see that Λ(t) ≤ c, for all t ≥ 0, as a consequence of Corollary 4.1, Lemma 6.2 and (H5). We now can write where we have used Lemma 6.2 once more. Integrating (6.5) with respect to time between s and t and taking account of Corollary 4.2 and of the bound Λ(t) ≤ c, for every t ≥ 0, we get the thesis.
Collecting the above results, for all z 0 ∈ B 0 and all t ≥ s ≥ 0 we have the bounds for every ω > 0.
Lemma 6.5. We have Proof. We differentiate (6.3) 1 and (6.3) 2 with respect to time and we set v := u 2t and q := p 2t . We obtain We multiply (6.25) 1 by v t + ǫv, (6.25) 2 by q t + ǫq, with ǫ > 0 to be chosen later, and we add the resulting equations. Performing the same kind of calculations done in the proof of Lemma 6.4, we get If we set we are thus led to dΛ dt + ǫΛ + Γ + ǫ 2 A 1/2 q 2 + q t 2 = (ψ ′′ (p 2 )p 2t , q 2 ) + 2θ(p t , q t ) + 2ǫθ(p t , q).(6.28) We now have by (H5) and Lemma 6.2, being z 0 ∈ B 0 . Therefore, from (6.28), (6.29) we deduce using Corollary 4.1 and Lemma 6.2 again in the last estimate. Now, it is not difficult to see, with the help of the Young inequality, that, for 0 < ǫ ≤ ǫ 0 with ǫ 0 small enough, and θ ≥ l, we have and Γ ≥ 0. From (6.30) we thus obtain dΛ dt + ǫΛ ≤ c p 2t 2 Λ + c (6.31) and the application of Lemma 2.2 to the differential inequality (6.31) (cf. (6.7)) yields the bound Λ(t) ≤ c for all t ≥ 0, and for all z 0 ∈ B 0 , which entails From this last bound we recover, with the help of (6.3) 1 and (6.3) 2 , the further controls Bu 2 (t) ≤ c and Ap 2 (t) ≤ c, from which we get the thesis.
We can now conclude the proof of the main result of this section.
Proof of Theorem 6.1. Let us put B 1 := {z ∈ H 1 : z 1 ≤ c} with c as in Lemma 6.5. Then, by using Lemma 6.4 and Lemma 6.5 we immediately get (6.1).
7. The exponential attractor. The global attractor is not always a nice object to describe the longterm dynamics of a system. Actually, the rate of convergence of the trajectories to the attractor is not controlled in general and, in some concrete cases, may be arbitrarily small (see, e.g., [13]). This means that, when only the existence of the global attractor is proved, it is in general very difficult, if not impossible, to know the time needed to stabilize the system. A more precise information on the convergence rate can be achieved when, for instance, the stationary solutions are hyperbolic. In that case, the global attractor (the so-called regular attractor in the terminology of A.V. Babin and M.I. Vishik [1]) is regular and exponential (in a sense that we shall precise below). For the regular attractors the rate of convergence can also be estimated in terms of the hyperbolicity contants of the equilibria, but, even in this situation, it is usually very difficult to estimate these constants for concrete equations. We also point out that the global attractor presents other defaults. Indeed, it is very difficult to express the convergence rate in terms of the physical parameters of the problem and, in addition, the global attractor may be sensitive to perturbations. In order to overcome all these difficulties Eden, Foias, Nicolaenko and Temam (see [4], [5]) introduced the notion of exponential attractor. This is a compact, positively invariant subset of the phase-space of finite fractal dimension that attracts bounded subsets of initial data exponentially fast (with a rate independent from the chosen subset of initial data) and it is more robust under perturbations. Unfortunately, contrary to the global attractor, the exponential attractor is not unique. However, if there exists an exponential attractor E, then the semigroup possesses a compact attracting set, and thus it has a global attractor A ⊂ E of finite fractal dimension, being dim F A ≤ dim F E. In spite of its lack of uniqueness, the exponential attractor can be considered a good compromise between the necessity of confining the longterm dynamics in a small set, and the necessity of having a satisfactory time control of the convergence of the trajectories. For a more extensive discussion and a review on recent results on exponential attractors for evolution equations we refer the reader to, e.g., [14].
In this section we prove that system (1.1) subject to Dirichlet-Neumann omogeneous boundary conditions possesses an exponential attractor which attracts all bounded subsets of H 0 . We first recall, for the reader's convenience, the definition of exponential attractor, which is a generalization of the definition in [4], [5] justified by the fact that we can prove that the exponential attractor has a basin of attraction coinciding with the whole phase-space (see, e.g., [15]).
Definition 7.1. A compact set E ⊂ H 0 is called an exponential attractor for the semigroup S(t) if the following conditions hold (i) E is positively invariant, that is, S(t)E ⊂ E for every t ≥ 0; (ii) E has finite fractal dimension, that is, dim F E < ∞; (iii) there exists an increasing function J : [0, +∞) → [0, +∞) and κ > 0 such that, for every R > 0 and for every set B ⊂ H 0 with sup z0∈B z 0 0 ≤ R there holds dist H0 (S(t)B, E) ≤ J(R)e −κt . (7.1) We are now ready to state the main result of this section.
Theorem 7.2. In the hypotheses (H3)-(H6) the semigroup S(t) on H 0 associated with problem P possesses an exponential attractor E.
In order to prove Theorem 7.2 we exploit the approach introduced in [6] by Efendiev, Miranville and Zelik, which allows the construction of the exponential attractor without the use of orthogonal projectors. For our purpose the abstract result we need is the following (see [15]) where S d and S c satisfy the conditions S d (z 2 ) − S d (z 1 ) 0 ≤ 1 8 z 2 − z 1 0 , ∀z 1 , z 2 ∈ X and S c (z 2 ) − S c (z 1 ) 1 ≤ C * z 2 − z 1 0 , ∀z 1 , z 2 ∈ X for some C * > 0.
Then there exists an invariant compact set E ⊂ X such that where N * is the minimum number of balls of radius 1 8C * of H 0 necessary to cover the unit ball of H 1 .
Besides Lemma 7.3 we recall another important abstract result which, in particular, shall be applied to show that the basin of attraction of the exponential attractor coincides with the whole phase-space. This result concerns the transitivity property of exponential attraction (see [7,Theorem 5.1]).