GLOBAL ATTRACTOR FOR A

. We prove the existence of a compact, ﬁnite dimensional, global attractor for a system of strongly coupled wave and plate equations with non-linear dissipation and forces. This kind of models describes ﬂuid-structure interactions. Though our main focus is on the composite system of two partial diﬀerential equations, the result achieved yields as well a new contribution to the asymptotic analysis of either (uncoupled) equation.


1.
Introduction. The mathematical model under consideration consists of a semilinear wave equation defined on a bounded domain Ω, which is strongly coupled with Berger's plate equation acting only on a part of the boundary of Ω. This kind of models, referred in the literature to as structural acoustic interactions, arise in the context of modeling gas pressure in an acoustic chamber which is surrounded by a combination of hard (rigid) and flexible walls. The pressure in the chamber is described by the solution to a wave equation, while vibrations of the flexible wall are described by the solution to a plate equation.
More precisely, let Ω ⊂ R n be a smooth bounded domain, n = 2, 3, with the boundary ∂Ω =: Γ = Γ 0 ∪ Γ 1 consisting of two open (in induced topology) connected disjoint parts Γ 0 and Γ 1 of positive measure. Γ 0 is flat and is referred to as the elastic wall, whose dynamics is described by the Berger plate (n = 3) or beam (n = 2) equation; for details on Berger model we refer to [15,Chap. 4] and to the literature quoted therein. The acoustic medium in the chamber Ω is described by a semilinear wave equation. Thus, we consider the following PDE system In the above system, g(s) and b(s) are non-decreasing functions describing the dissipation effects in the model, while the term f (z) represents a nonlinear force acting on the wave component; ν is the outer normal vector, α and β are positive constants; the parameter κ ≥ 0 has been introduced in order to cover also the case of non-interacting wave and plate equations (κ = 0). The part Γ 1 of the boundary describes a rigid (hard) wall, while Γ 0 is a flexible wall where the coupling with the plate equation takes place. The boundary term βκz t | Γ0 describes back pressure exercised by the acoustic medium on the wall. The real parameter Q describes in-plane forces applied to the plate, while p 0 ∈ L 2 (Ω) accounts for transversal forces. The described interactive system involves a coupling between n-and (n− 1)dimensional manifolds, and as such is of hybrid type [35]. Structural acoustic models are well known in both the physical and mathematical literature and go back to the canonical models considered in [6,36,28]. More recently, these models were studied in the context of control theory, where problems such as the active control of pressure and vibrations by means of actuators placed on the flexible wall, or stabilization and controllability of the overall structure become issues of focal interest. There is a very large literature devoted to this topic; the reader is referred to the monograph [32], which also provides a rather comprehensive overview of related works. More recent contributions include [2,3,13,26,30], where questions of exact controllability or uniform stability are dealt with for interactions of wave/Kirchhoff plates [3], wave/Reissner-Mindlin plates [26] or wave/shell models [13], respectively. We also mention the papers [7,8,9,14] which consider the coupled system of a linear wave equation (in Ω = R 3 + := {(x 1 , x 2 , x 3 ) : x 3 > 0}) and von Karman equations (in Γ 0 ). This case corresponds to nonlinear aeroelastic plate problem in a flow of gas.
In contrast, in the present paper we study the long-time behavior of structures subject to nonlinear external or internal excitations. This type of issues are dealt with within the framework of dynamical systems. Hence, central questions to be discussed are the existence of global attractors and their properties. Our choice of a semilinear wave equation and Berger's plate model, both with nonlinear damping, beside being of great physical relevance, is also suggested by mathematical considerations. These two models provide canonical examples of relatively simple hyperbolic-like PDE systems, yet they exhibit the main technical difficulties encountered in proving existence of attractors, or asserting their regularity or finitedimensionality in the presence of a nonlinear damping. In fact, the intrinsic hyperbolic (hyperbolic-like) character of the dynamics involved makes the study of long time behavior of the corresponding models somewhat challenging. This is related to the fact that the unstable part of the free models is infinite-dimensional (unlike the case of parabolic-like equations). This puts a strong demand on the damping which must control an "infinite dimensional spectrum". In the case of linear damping, this has been dealt with rather successfully [4,15,27,40]. However, the presence of nonlinear damping along with the intrinsic non-differentiability of the flow on the phase space has been recognized in the literature as a source of main difficulties.
In recent years there has been a steady progress in this area, particularly in the context of wave equations [25,33,37], and novel techniques have been developed with an eye to hyperbolic-like dynamics exemplified by second order evolutions. The recent work by the second and third authors [18,19,20] provides a comprehensive account of new abstract results, along with the analysis of relevant PDE examples such as waves and plates equations with nonlinear damping and critical (i.e. noncompact) nonlinear terms.
The present work takes advantage of the new methods developed or reported in [20], which are applied in the context of interactive structures. This way we not only achieve sharp results for structural acoustic interactions, but also improve some of the previous results available for the uncoupled models.
In this paper we aim to show the existence and to study the properties of a global attractor for problem (1).
Our first main result, Theorem 3.2, states the existence of a global attractor for problem (1) under rather general conditions on the nonlinear functions g, f and b; see Assumption 3.1. In particular, it is not assumed that the damping functions g and b are (i) differentiable, nor (ii) strictly increasing. The implications of this result for the uncoupled wave and plate equations are also new.
Our second main result, Theorem 3.4, deals with the dimension and smoothness of the global attractor. It requires additional (structural) hypotheses concerning the damping functions g and b and the nonlinear force f ; see Assumption 3.3. In particular, strong monotonicity of g and b will be assumed. In the case κ = 0, this result is in accordance with the results previously established in [18], [19,Chap. 5] and [22] for the wave equation, and in [20,Chap. 7] for Berger model. The statements pertaining to either uncoupled equation are given explicitly in Corollary 3.5 and Corollary 3.6, respectively.
The paper is organized as follows. Section 2 contains some background material: we give here the abstract formulation of problem (1), we then state Theorem 2.3 on well-posedness of the PDE problem (whose proof is left to an Appendix), and describe the properties of suitable energy functionals and stationary solutions. In Section 3 we state our main results, namely Theorem 3.2 on the existence of a compact global attractor for system (1), and Theorem 3.4 on the attractor's dimension and smoothness. Their Corollaries 3.5 and 3.6, followed by detailed remarks, are also given here. In Section 4 we develop some preliminary tool for the proofs of Theorems 3.2 and 3.4. Section 5 contains the proof of asymptotic smoothness of the dynamical system generated by (1): this property is crucial in order to show existence of a global attractor established in Theorem 3.2. In Section 6 we establish a so-called stabilizability estimate, which will play a key role in the proof of Theorem 3.4 in Section 7.
2. Preliminaries. The notation below is largely standard within the literature. For the reader's convenience, we just recall that the symbols || · || O and (·, ·) O denote the norm and the inner product in L 2 (O). The subscripts in (·, ·) O and || · || O will be often omitted when apparent from the context. We denote || · || σ,O the norm in the L 2 -based Sobolev space H σ (O). Here we will have either O = Ω or O = Γ 0 . We also denote by Basic assumption. We shall impose the following basic assumptions on the nonlinear functions g and f which affect the wave component of the system and on the damping function b in the plate equation.
• g ∈ C(R) is a non-decreasing function, g(0) = 0, and there exists a constant C > 0 such that where 1 ≤ p ≤ 5 when n = 3, while 1 ≤ p < ∞ for n = 2. • f ∈ Lip loc (R), and there exists a positive constant M such that where q ≤ 2 when n = 3, and q < ∞ for n = 2. Moreover, the following dissipativity condition holds true: • b ∈ C(R) is a non-decreasing function such that b(0) = 0.

Abstract formulation.
We find convenient to represent the PDE system (1) in an abstract-semigroup form. In fact, we shall see that the resulting formulation is a special case of a general nonlinear structural acoustic model given in [32,Sect. 2.6].
In order to accomplish this we introduce the following spaces and operators. Let A : D(A) ⊂ L 2 (Ω) → L 2 (Ω) be the positive self-adjoint operator defined by where µ > 0 is given by (4). Next, let N 0 be the Neumann map from L 2 (Γ 0 ) to L 2 (Ω), defined by It is well known (see, e.g., [34,Chap.3]) that In particular, we have that It is also well known that by Green's second theorem, the following trace result holds true (see, e.g., [34]): where N * 0 : L 2 (Ω) → L 2 (Γ 0 ) is the adjoint of N 0 . The validity of (6) may be extended to all h ∈ H 1 (Ω) ≡ D(A 1/2 ), as D(A) is dense in D(A 1/2 ). Regarding the plate model, let A : D(A) ⊂ L 2 (Γ 0 ) → L 2 (Γ 0 ) be the positive, self-adjoint operator defined by The fractional powers of A are well defined and we have, in particular, . By using the above dynamic operators, the coupled PDE problem (1) can be rewritten as the following abstract second order system, which is a particular case of the one studied in [32,Sect. 2.6]: where we have introduced the operators Regarding the nonlinear force terms we have that where ′ stands for the Fréchet derivative in an appropriate space. It readily follows from (4) that for some positive constants δ f and M f . Similarly, we have that The phase spaces Y 1 for the wave component [z, z t ] and Y 2 for the plate component [v, v t ] of system (7) are given, respectively, by: which will be supplemented with the following norm and with the corresponding inner product.

118
FRANCESCA BUCCI, IGOR CHUESHOV AND IRENA LASIECKA Notice that an important consequence of Assumption 2.1 and of criticality of the parameter q is that the nonlinear operators F 1 and F 2 are locally Lipschitz continuous from H 1 (Ω) into L 2 (Ω) and from H 2 (Γ 0 ) ∩ H 1 0 (Γ 0 ) into L 2 (Γ 0 ), respectively, that is where C(ρ) denotes a function which is bounded for bounded arguments. It is important to emphasize that while the operators are bounded on the respective spaces, they are not compact. This fact justifies the notion of "criticality" for the parameter q and for the nonlinear terms F 1 and F 2 . The natural (nonlinear) energy functions associated with the solutions to the uncoupled wave and plate models are given, respectively, by where we have set Since both energy functionals in (14) may be negative, it is convenient to introduce the following positive energy functions where M f is the constant in (9). Finally, we introduce the total energy , v t (t)) of the system, namely whose positive part is given by It is easy to see from the structure of the energy functionals and in view of (9) and (10) that for any α, β > 0 there exist positive constants c, C, and M 0 such that where E and E are the energies defined in (16) and (17). 2.3. Well-posedness. In this subsection we study well-posedness of problem (1).
Since the corresponding abstract system (7) is a special case of a general abstract model studied in [32, Sect. 2.6], local and global existence results for the solutions can be deduced from Theorem 2.6.1 and Theorem 2.6.2 in [32]. These results are based on the theory of monotone operators; see [5], [11], or [38]. In order to make our statements precise, we need to introduce the concepts of strong and generalized solutions.
which satisfy the initial conditions (7c) is said to be (S) a strong solution to problem (7) on the interval [0, T ], iff • for any 0 < a < b < T one has The main result concerning well-posedness of problem (1) is the following assertion.
, v t (t)) which depends continuously on initial data. This solution satisfies the energy inequality with the total energy E(t) given by (16). Moreover, if initial data y 0 are such that , then there exists a unique strong solution y(t) satisfying the energy identity: Both strong and generalized solutions satisfy the inequality The existence and uniqueness of generalized solutions asserted in Theorem 2.3 follows from Theorem 2.6.2 in [32]. To see this, it is sufficient to notice that the conditions in Assumption 2.1 imply that (i) the operators F 1 and F 2 are locally Lipschitz continuous in the respective spaces (see (13)), with appropriate a-priori bounds, and (ii) the damping functions modeled by g and b satisfy the monotonicity and continuity properties required in [32,Sect. 2.6]. However, for the sake of completeness and since [32] is focused on nonlinear boundary damping, which requires additional technicalities, we shall provide in the Appendix a self-contained proof, tailored for the specific problem under investigation. Remark 1. The existence of generalized solutions established in Theorem 2.3 is obtained by using the theory of nonlinear semigroups. As such, these solutions are defined as strong limits of regular (strong) solutions, as in the part (G) of Definition 2.2. This does not automatically imply that generalized solutions satisfy a variational equality. However, in the present paper, the regularity of g and f enable us to compute appropriate limits and to obtain the variational form stated below. Indeed, using the same argument as in [21] one can prove that any generalized solution (7) is also weak, i.e. it satisfies the following (variational) relations: for any φ ∈ H 1 (Ω) and ψ ∈ H 2 (Γ 0 ) ∩ H 1 0 (Γ 0 ) in the sense of distributions. Theorem 2.3 makes it possible to define a dynamical system (Y, S t ) with the phase space Y given by (11) and with the evolution operator S t : Y → Y given by the relation is a generalized solution to (7). Moreover, the monotonicity of the damping operators D and B, combined with the estimate in (13) and the boundedness property given by (22) imply (by a pretty routine argument) that the semi-flow S t is locally Lipschitz on Y . Moreover, there exists a > 0 and ω(ρ) > 0 such that 2.4. Energy functionals and stationary solutions. We conclude this section by discussing several properties of the energy functionals and stationary solutions. It follows from (18) This, in turn, implies that there exists R * > 0 such that the set is a non-empty bounded set in Y for all R ≥ R * . Moreover any bounded set B ⊂ Y is contained in W R for some R and, as it follows from (21), the set W R is invariant with respect to the semi-flow S t , i.e. S t W R ⊂ W R for all t > 0. Thus we can consider the restriction (W R , S t ) of the dynamical system (Y, S t ) on W R , R ≥ R * . We introduce next the set of stationary points of S t denoted by N , Using the properties of the potentials Π and Φ given by (8) and (10) one can easily prove the following assertion.
Lemma 2.4. Under Assumption 2.1 the set N of stationary points for the semiflow S t generated by equations (7) is a closed bounded set in Y , and hence there exists R * * ≥ R * such that N ⊂ W R for every R ≥ R * * .
Later we will also need the notion of unstable manifold M u (N ) emanating from the set N , which is defined as the set of all W ∈ Y such that there exists a full trajectory γ = {W (t) : t ∈ R} with the properties We finally recall that a continuous curve 3. The statement of main results. The goal of the present paper is to show the existence of a global attractor for the dynamical system generated by problem (1), and to study its properties.
Let us recall (cf. [4,15,27,40]) that a global attractor for a dynamical system (X, S t ) on a complete metric space X is a closed bounded set A in X which is invariant (i.e. S t A = A for any t > 0) and uniformly attracting, i.e. lim t→+∞ sup y∈B dist X {S t y, A} = 0 for any bounded set B ⊂ X.
The fractal dimension dim f M of a compact set M is defined by where N (M, ε) is the minimal number of closed sets of diameter 2ε which cover the set M .
To prove the existence of a global attractor for problem (7) we need additional hypotheses concerning the damping functions g and b.
Alternatively, (26) and (27)  In particular, this means that Assumption 3.1 allows the damping functions g and b to be constants on some closed finite intervals which are away from zero.
Our first main result is the following theorem.
Theorem 3.2. Under Assumption 3.1 the dynamical system (Y, S t ) generated by problem (7) has a compact global attractor A which coincides with the unstable manifold M u (N ) emanating from the set N of stationary points for S t , A ≡ M u (N ).
To state our second main result we need additional hypotheses.
Remark 3. If g, b ∈ C 1 (R), then (30) and (31) are equivalent to the requirements and for some constants m ′ , M ′ > 0 and m ′ 1 , M ′ 1 > 0. Moreover, one can see that the inequality on the right hand side of (32) holds true if we assume that for some m ′′ > 0. We also note that the second requirement in (34) follows from the first one if 1 ≤ p ≤ 1 + 2σ. The same is true in the case of (33). In addition, the condition (33) allows also an exponential behavior for b(s), e.g., |b(s)| ∼ e α|s| as |s| → ∞ for some α > 0.
Our second main result is the following theorem. 1. The attractor A has a finite fractal dimension.

The attractor A is a bounded set in the space
in the case n = 3 and 3 < p ≤ 5, where W 2 6/p (Ω) is the L 6/p -based second order Sobolev space, and in the space 3. There exists a fractal exponential attractor A exp for (Y, S t ) (whose dimension is finite in some extended space) provided that b(s) is polynomially bounded at infinity.
The concept of fractal exponential attractor has been introduced in [23]. Let us recall from [23] that a compact set A exp ⊂ X is said to be an inertial set (or a fractal exponential attractor) for a dynamical system (X, S t ) iff (i) A exp is a positively invariant set of finite fractal dimension and (ii) for every bounded set D ⊂ X there exist positive constants C D and γ D such that Fractal exponential attractors have been intensively studied by many authors. We refer to [24] for a recent survey with focus on second order evolution equations in the case of linear damping, and to [19,20] for results for nonlinearly damped models.
In the case κ = 0 the above theorems result in the following two assertions. for (Y 1 , S 1 t ). In contrast to most results available in the literature (see, e.g., [18], [19,Chap. 5], [39] and the references therein) our Corollary 3.5 deals with the wave dynamics subject to Neumann boundary conditions and-which is more important-does not require differentiability and strict monotonicity of the damping g for the existence of a global attractor. Furthermore, the present result does not require either subcriticality (unlike [39]) or superlinearity (unlike [19, Chap. 5]) of the damping function g. Since the same method can be also applied in the case of Dirichlet boundary conditions, we achieve as well a generalization of the results given in [19,Chap. 5] and [39]. We finally stress that, in contrast to [18] and [19,Chap. 5], our result on the existence of a fractal exponential attractor requires neither additional geometrical assumptions concerning the set N 1 of equilibria, nor large damping hypothesis.
In the case of Berger's model, Theorems 3.2 and 3.4 yield the following result.

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FRANCESCA BUCCI, IGOR CHUESHOV AND IRENA LASIECKA Corollary 3.6. Suppose that b and p 0 satisfy the conditions in Assumption 2.1 and the inequality (27). Then the dynamical system (Y 2 , S 2 t ) generated by problem possesses a compact global attractor A 2 ≡ M u (N 2 ), where N 2 is the set of equilibria for (36). If, in addition, condition (31) holds true, then the attractor A 2 is a bounded set in D(A) × D(A 1/2 ) and has a finite fractal dimension. Moreover, the system (Y 2 , S 2 t ) possesses a fractal exponential attractor A exp 2 provided that b(s) is polynomially bounded at infinity.
In the case when b ∈ C 1 (R), the existence of a global attractor for (36) was established in [19,Chap. 7] under the additional hypotheses that b(s) is strictly increasing and b ′ (s) ≥ m > 0 for |s| ≥ 1. As for dimension and smoothness of the attractor, in the special case when b ∈ C 1 (R), Corollary 3.6 requires the same hypotheses as in [20,Chap. 7]. Instead, the issue of existence of a fractal exponential attractor for Berger's model with nonlinear damping had not been explored before.

Main inequality. The key ingredient for the proofs of both Theorems 3.2 and 3.4 is the following Proposition.
Proposition 4.1. Let Assumption 2.1 hold. Assume that y 1 , y 2 ∈ W R for some R > R * , where W R is defined by (25) and denote Let z(t) := h(t)−ζ(t) and v(t) := u(t)−w(t). Then, there exists T 0 > 0 and positive constants c 0 , c 1 and c 2 (R) independent of T such that for every T ≥ T 0 where E 0 (t) = βE 0 z (t) + αE 0 v (t) with E 0 z (t) and E 0 v (t) given by (15). We also used the notations and As we shall see, the inequality (37) established in Proposition 4.1 provides a common first step for the proof of all the statements of Theorem 3.2 and Theorem 3.4. This inequality represents equipartition of the energy: the potential energy is reconstructed from the kinetic energy and the nonlinear quantities entering the equation. Eventually, these quantities will need to be absorbed ('modulo' lower order terms) by the dampings. The realization of this step depends heavily on the assumptions imposed on the model, and hence the argument used in the proof of compactness will be different from the one given for the proof of finite-dimensionality and/or regularity. Proof.
Step 1 (Energy identity). Without loss of generality, we can assume that (h(t), u(t)) and (ζ(t), w(t)) are strong solutions. By the invariance of W R and in view of relation (18) there exists a constant C R > 0 such that where E 0 z (t) and E 0 v (t) denote the corresponding (free) energies as in (15). We establish first an energy type equality regarding E 0 (t) = βE 0 where G T t (z) and G T t (v) are given by (38), while F 1 (z) and F 2 (z) are defined by (41).
Proof. Writing down the equations satisfied by h = ζ + z, u = w + v, ζ and w (see (7)), it is elementary to derive the following system of coupled equations: with F i defined in (41). Next, by standard energy methods we obtain Combining (45) with (46) we immediately see that (43) holds true.
Step 2. (Reconstruction of the energy integral) We return to the coupled system (44) satisfied by (z, v). We multiply equation (44a) by z, and integrate between 0 and T , thereby obtaining where H T 0 (z) is defined in (39). It is clear from (13) and (42) that |(F 1 (z), z)| ≤ C R A 1/2 z z .
Using (5) with ε = 1/4 we have that Regarding the plate component, by using the bounds in (13) and (42) we obtain where H T 0 (v) is defined in (39). Integrating by parts in time and using the standard form of the trace theorem it is easy to see that for every ε > 0. Consequently, summing up (47) with (48) and using (49), we get On the other hand, it follows from Lemma 4.2 that and (52) Therefore, combining (52) with (50) and (51), it is readily shown that (37) holds true, provided that T is sufficiently large. This concludes the proof of Proposition 4.1.

5.
Asymptotic smoothness and proof of Theorem 3.2. In this section we will show that the semi-flow S t generated by the PDE system (1) is asymptotically smooth. This property is critical for proving existence of global attractors (see, e.g., [4,15,27,40]). We recall (see, e.g., [27]) that a dynamical system (X, S t ) is said to be asymptotically smooth iff for any bounded set B in X such that S t B ⊂ B for t > 0 there exists a compact set K in the closure B of B, such that lim t→+∞ sup y∈B dist X {S t y, K} = 0.
Our main result in this section is the following assertion.
Theorem 5.1. Let Assumption 3.1 hold. Then the dynamical system (Y, S t ) generated by the PDE problem (1) is asymptotically smooth.
In order to prove Theorem 5.1 we shall invoke a compactness criterion due to [29,Thm. 2], which is recalled below in an abstract version formulated and used in [21] (see also [20,Chap.2]).

Proposition 5.2 ([21]
). Let (X, S t ) be a dynamical system on a complete metric space X endowed with a metric d. Assume that for any bounded positively invariant set B in X and for any ε > 0 there exists T = T (ε, B) such that where Ψ ε,B,T (y 1 , y 2 ) is a function defined on B × B such that for every sequence {y n } n in B. Then (X, S t ) is an asymptotically smooth dynamical system.
In the proof of Theorem 5.1 we shall make use of further inequalities which are a standard tool for proving the absorption property; see [20].
The second statement follows trivially from the Sobolev's embedding

and the obvious inequality
for further details we refer to [18,19]).
Proof of Theorem 5.1. Since any bounded positively invariant set belongs to W R for some R > R * , where W R is defined by (25), it is sufficient to consider the case B = W R for every R > R * only. Let y 1 , y 2 ∈ W R . Below we use the same notations as in Proposition 4.1. Namely, we denote the solutions corresponding to initial data y 1 and y 2 , respectively, by (h(t), h t (t), u(t), u t (t)) := S t y 1 , (ζ(t), ζ t (t), w(t), w t (t)) := S t y 2 , and set z(t) := h(t) − ζ(t) and v(t) := u(t) − w(t).
Now we establish the key estimate for the proof of Theorem 5.1.
Proposition 5.4. Let the assumptions of Theorem 5.1 be in force. Then, given ε > 0 and T > T 0 there exists constants C ε (R) and C(R, T ) such that where E 0 (t) := βE 0 Proof. It follows from the energy inequality (19) that where crucially C R does not depend on t.
Let H t s (z) and H t s (v) be given by (39). By using Lemma 5.3, the estimates (57) and the fact that A 1/2 z ≤ C R , we obtain Using now the inequalities in Assumption 3.1 and once again the uniform estimates (57), one can see that Taking first t = 0 in (43) and using the fact that E 0 (0) ≤ C R , we get Therefore, (56) follows from Proposition 4.1 and the estimates (58), (59) and (60).
We are now in a position to complete the proof of Theorem 5.1. It follows from Proposition 5.4 that given ε > 0 there exists T = T (ε) > 1 such that for initial data y 1 , y 2 ∈ B we have where Ψ T (z, v) is given by (40). Thus, in order to establish asymptotic smoothness for the dynamical system under investigation, we aim to invoke Proposition 5.2, which will allow us to conclude the proof of Theorem 5.1. Hence, what we need to prove is the validity of the sequential limits (53) for Ψ ε,B,T defined by (61). To do that, we shall use similar arguments as in the completion of the proof of Theorem 3.1 in [21]. Some computations are given for the reader's convenience. Let (h n , u n ) n be a sequence of solutions to the PDE system (1) corresponding to initial data y 0,n := (h 0,n , h 1,n , u 0,n , u 1,n ) in W R ⊂ Y . Since the compactness condition in (53) deals with lower limits, it is sufficient to establish (53) for some subsequence of (y 0,n ) n . Therefore we can assume that as n, m → ∞, for some δ, η > 0, where we have set z n,m (t) = h n (t) − h m (t) and v n,m (t) = u n (t) − u m (t). The above convergence statement implies that lot(z n,m , v n,m ) → 0. Therefore, in view of (61), we must show that is continuous from L q+2 (Ω) into L 1 (Ω). Therefore, since H 1−δ (Ω) ⊂ L q+2 (Ω) for some δ > 0, the potential Π defined in (8) In order to estimate the term C n,m in (65) we first note that by (13) sup (67) From (62) and (63) we also have that h n → h strongly in L 2 (0, T ; L 2 (Ω)), and hence almost everywhere (along a subsequence). Since F 1 is continuous, we also have that F 1 (h n ) → F 1 (h), a.e. in (0, T ) × Ω. This allows to recover the limit in (67) and claim that F 0 = F 1 (h). Thus, letting m → ∞ first and n → ∞ next, we obtain for term C n,m : .
Recalling (40), and taking into account (70) and (71)  6. Stabilizability estimate. In this section we aim to develop some analytic tool which will enable us to apply the abstract results presented in [18] and [19,20] for the proof of Theorem 3.4. The crucial analytic tool will be a suitable so-called stabilizability estimate. Let us preliminarily record two key estimates, whose proof can be found in [18,19]. Lemma 6.1 ([18,19]). Under Assumptions 3.3, the following statements are valid.
The following Theorem provides a "stabilizability" inequality, which plays a key role in the proofs of both finite-dimensionality and regularity of attractors. This inequality shows that the difference of two trajectories can be exponentially stabilized to a compact (smooth) set. Theorem 6.2 (Stabilizability estimate). Let Assumption 3.3 hold. Then there exist positive constants C 1 , C 2 and ω depending on R such that for any y 1 , y 2 ∈ W R the following estimate holds true: where Above, we have used the notation (h(t), h t (t), u(t), u t (t)) := S t y 1 , (ζ(t), ζ t (t), w(t), w t (t)) := S t y 2 .
Proof. Let y 1 , y 2 ∈ W R be given. We consider the solutions S t y 1 and S t y 2 and introduce, as previously in the paper, z = h − ζ, v = u − w. We recall that for these solutions the bounds (42) and (57) hold true. We begin with the following critical assertion.
The idea behind the estimates in Lemma 6.3 is to exhibit explicitly the kinetic energy ||h t || 2 + ||ζ t || 2 , which is L 1 (R) (see (81) below). As such, this function may play a role of a small parameter for large |t|. As we shall see, this small "parameter" will allow to dispense with unnecessary assumptions such as subcritical damping parameter p, or large damping parameter in front of g(z t ) (assumptions used in the previous treatments; see [18] and the references therein).
Regarding the latter estimate (76b), one can readily check that with Σ(τ ) : For the above term we have that which inserted in (80) gives (76b), as desired.
By the lower bounds in (30) and (31) we have that where G T 0 (z) and G T 0 (v) are given by (38), and also Returning to Proposition 4.1 and using Lemma 6.3, we obtain for Ψ T (z, v) in (40) the estimate for every ε > 0, where E 0 (t) is the same as in Proposition 4.1 and with lot T (z, v) defined by (75) and D h,ζ (t) given in (82). The next assertion is a direct consequence of Lemma 6.1.
Lemma 6.4. Under Assumption 3.3, the following estimate holds true with arbitrarily small ε > 0: where G T 0 (z), G T 0 (v) are defined in (38), and Ξ T (z, v) is given by (84). Proof. It is sufficient to apply the estimates recalled in Lemma 6.1. In fact, by using (72) we readily obtain that given ε > 0, there exists a positive constant C ε such that Similarly, using both (73) and the bounds (57) we have that . The two estimates above immediately give (85), as desired.
with T defined by (92) and C given by As a consequence of (13), the nonlinear term C is locally Lipschitz on the phase space Y . Thus, according to Theorem 7.2 in [16] a local existence result will follow once we prove that T is a maximal accretive operator; see [11,5].
where we have used (11) and (12) to compute the inner product, and the monotonicity properties of g and b to obtain the latest inequality.
Maximality. As it is well known (see, e.g., [38, p. 18]), in order to show that T is maximal, we only need to prove that R(I + T ) = Y . We can do it in the same way as in [1,12]. Given h = [ϕ 1 , ϕ 2 , ψ 1 , ψ 2 ] T ∈ Y , we seek to solve the equation (I + T )y = h, which explicitly reads as Eliminating z 1 and v 1 we arrive to the equation with L and G given by Let us observe that the right hand side of (96) belongs to the dual space Y ′ of Y := D(A 1/2 ) × D(A 1/2 ) with respect to the duality ·, · Y ′ ,Y given by Moreover, we have in particular that L ∈ L(Y, Y ′ ) while G : Y → Y ′ is a monotone hemicontinuous operator by Assumption 2.1. The above decomposition is useful to show the following result.
Lemma A.1. Let L and G be the operators defined in (97). Then R(L + G) ≡ Y ′ .
Proof. By [5, Corollary 1.3, p. 48], it is necessary to verify the following properties: (i) L is a monotone, hemicontinuous operator from Y to Y ′ , (ii) G is a maximal monotone operator from Y to Y ′ , and (iii) L + G is coercive.
We first compute Y . This shows that L is coercive, hence as a linear operator it is a monotone hemicontinuous operator, and thus (i) holds true. Moreover, since G is monotone, then L + G is coercive, as well, so that (iii) is satisfied. It remains to be shown that G is maximal monotone as a mapping from Y to Y ′ . Since g 1 (s) := g(s) + s is increasing, then g 1 = ∂Φ(·), as a mapping from D(A 1/2 ) to [D(A 1/2 )] ′ , where Φ is some proper, convex, lower semi-continuous functional on D(A 1/2 ) and ∂Φ denotes the subgradient of Φ (cf. [10, p. 37]). The same property holds for b (in proper spaces). Thus, we can invoke [5, Theorem 2.1, p. 54] to obtain that G is a maximal monotone operator from Y to Y ′ , as desired.
Step 2. Energy inequalities/equalities. To derive the energy identity (20) for strong solutions on the existence time interval a standard procedure applies (see, e.g., [12] or Lemma 4.2 above), provided boundedness of the damping operator D(z t ) and B(v t ) as a map from D(A 1/2 ) (resp D(A 1/2 )) into the duals [D(A 1/2 )] ′ (resp [D(A 1/2 ))] ′ ) is ascertained. The latter follows from criticality of the parameter p (in the case of D) and from Sobolev's embedding D(A 1/2 ) ⊂ C(Γ 0 ) (in the case of B). In fact, the damping B is smoother than required, since B(D(A 1/2 )) ⊂ C(Γ 0 ). To establish the energy inequality (19) for generalized solutions, we only need to justify the limit transition (from strong to generalized solutions) in the damping terms. This can be done exploiting the properties of L 2 -convergence and appropriate approximations of the damping functions g and b (the argument is the same as in [21]). Finally, the crucial property (21) that the energy is non-increasing immediately follows from the energy inequality (19). The upper bound (22) is obtained combining (21) with (18).
Step 3. Global existence. It follows from (22) that the solution cannot blow up in finite time. Therefore the same argument in Theorem 7.2 from [16] (see also [19, Chap.1]) shows global existence for both strong and generalized solutions.