FINITE DIMENSIONAL SMOOTH ATTRACTOR FOR THE BERGER PLATE WITH DISSIPATION ACTING ON A PORTION OF THE BOUNDARY

. We consider a (nonlinear) Berger plate in the absence of rotational inertia acted upon by nonlinear boundary dissipation. We take the boundary to have two disjoint components: a clamped (inactive) portion and a controlled portion where the feedback is active via a hinged-type condition. We emphasize the damping acts only in one boundary condition on a portion of the boundary . In [24] this type of boundary damping was considered for a Berger plate on the whole boundary and shown to yield the existence of a compact global attractor. In this work we address the issues arising from damping active only on a portion of the boundary, including deriving a necessary trace estimate for (∆ u ) (cid:12)(cid:12) (cid:0) 0 and eliminating a geometric condition in [24] which was utilized on the damped portion of the boundary. Additionally, we use recent techniques in the asymptotic behavior of hyperbolic-like dynamical systems [11, 18] involving a \stabilizability" estimate to show that the compact global attractor has (cid:12)nite fractal dimension and exhibits additional regularity beyond that of the state space (for (cid:12)nite energy solutions).


Introduction.
In this treatment, we study a partial differential equation (PDE) model that accounts for the nonlinear vibrations of a thin, elastic plate-in the sense of large deflections [6,7,21]. The physical derivation of this plate model, as well as a detailed discussion of the validity of the model, can be found in [24]. The particular interest of this treatment is to analyze Berger's plate model from the point of view of long-time behavior. We are interested in global attractors and their structural properties (such as finite dimensionality and regularity) in the presence of nonlinear dissipation acting via moments on a portion of the boundary (the remaining, disjoint portion of the boundary will be taken to be clamped). To obtain asymptotic smoothness of the dynamics (see the Appendix for a definition) a standard geometric condition on the undamped portion of the boundary will be in force.

Model.
Let Ω ⊂ R 2 be a bounded domain with smooth boundary Γ. We assume the plate has negligible thickness [15,17,18,21,30]. We take Γ = Γ 0 ⊔ Γ 1 , where ⊔ denotes disjoint union and Γ 0 ̸ = ∅; Γ 0 will be considered the undamped portion of the boundary, while Γ 1 is the active portion of the boundary. Thus, invoking the hypothesis of (i) finite elasticity, (ii) assuming filaments of the plate remain perpendicular to the central plane of the plate throughout deflection, and (iii) neglecting in-plane accelerations [30], we obtain: with clamped-hinged boundary conditions and dissipation acting via moments [15,17,18,30], denoted (CHD)-clamped-hinged dissipation: The damping function g(·) ∈ C 1 (R) is taken to be monotone increasing, and g(0) = 0. The Berger nonlinearity is given by This nonlinearity is a simplification of the scalar von Karman nonlinearity [18,21,30] which is obtained by assuming the second strain-invariant to be negligible [6,24]. Doing so is generally considered valid for clamped and hinged type plate boundary conditions [43] (and references therein). The source function p ∈ L 2 (Ω) represents static pressure differences across the surface of the thin plate. The parameter Υ > 0 is a physical parameter measuring the strength of the effect of stretching on bending, and γ corresponds to in-plane tension (γ < 0) or compression (γ > 0) [6]. In this treatment (i) we consider the non-dissipative case γ ≥ 0, as is by now standard in treatments of the Berger plate [13,16], and (ii) we normalize Υ = 1.
1.2. General discussion of previous work. The asymptotic-in-time behavior of the hyperbolic-like dynamics has been a topic of immense study over the past 30 years [4,5,12,14,22,23,27,33,38,41]. We recommend the resource [18]focused on von Karman plate dynamics-for a modern and comprehensive study of the long-time behavior (attractors and related topics) of second order (in time) abstract equations. In addition to its expansive coverage of von Karman plates, the monograph has a detailed exposition of the abstract theory of nonlinear dynamical systems, compact global attractors, and further asymptotic properties of solutions to hyperbolic-like PDEs, as well as key references for each of the principal topics previously mentioned. We also mention [15,17], which are precursors to much of the work in [18] for boundary dissipation acting on a nonlinear plate.
In this treatment, the focus is the existence and structural properties of a compact global attractor for the non-rotational Berger plate (1) taken with (CHD) conditions. We base our approach on the modern analysis of dissipative dynamical systems [11,18]. In such an approach, we firstly show that the dynamical system generated by generalized (semigroup) solutions to (1), taken with (CHD) conditions, is (ultimately) dissipative; i.e., it has a bounded absorbing ball in the state space. (This dissipative property of the dynamical system does not require any geometric assumptions on Γ.) To accomplish this, we use a multiplier method with the nonlinear energy functional, as developed in [14] (for the semilinear wave equation), and utilized later in [24] for the problem at hand taken with the entire boundary being damped. We then prove that the given dynamical system is asymptotically smooth, by using the well-known iterated compensated compactness criterion from [18] (first developed in [29]); this requires a standard geometric assumption on the uncontrolled portion of the boundary. After showing the existence of the compact attractor for the dynamics, we show that the attractor has finite fractal dimension in the state space, and that elements taken from this attractor have additional smoothness. This relies on modern techniques which utilize a "stabilizability" estimate (see [18]).
The natural energy for linear plate dynamics is given by the sum of the potential and kinetic energies The dynamics evolve in the state space We will also critically use the following nonlinear energies associated to equation in (1): where the Π term represents the non-dissipative and nonlinear portion of the energy: As in [18,Lemma 1.5.4], for any u ∈ (H 2 Γ0 ∩ H 1 0 )(Ω), 0 < η ≤ 2 and ϵ > 0, This yields the following crucial fact: for all u ∈ (H 2 Γ0 ∩ H 1 0 )(Ω). From (5), we have the bounds for some c 0 , c 1 , C > 0 depending on p and γ. Accordingly, we introduce more notation for the study of long-time behavior (following [14]): where M = M (ϵ, γ, p) is the constant given in (5).
In the context of this treatment, we are interested in non-homogeneous boundary conditions where the dissipation mechanism is active only on a portion of the boundary. We consider generalized (nonlinear semigroup) solutions; these are strong limits of strong solutions in the energy topology which satisfy an integral formulation of (1). For more details on the notion of solution, and for precise definitions of strong and weak solutions, see [18] and [24]. Although the well-posedness of Berger plate equation has been known for some time in the case of homogeneous boundary conditions, it is more recent [24] under non-homogeneous boundary conditions when the damping is active via moments on the whole boundary. The approach relies on adaptations of the abstract results in [18] for second order (in time) evolutions with nonlinear damping. This result can easily be adapted to our case after some minor modifications of the abstract functions and operators defined in [24,Section 4.1]. But in order to formulate this result, we need the following assumption on the damping function (linear-like away from the origin): Here we note that-for well-posedness-one can make weaker assumptions on the damping mechanism g(s); e.g., g(s) could exhibit arbitrary polynomial growth. However, for our subsequent results for long time behavior it is necessary that g(s) satisfy the stronger assumption above which is also utilized in [14,18].
We now give the well-posedness result: Theorem 2.1. Let Assumption 1 hold. With reference to (1) taken with (CHD) boundary conditions and with initial data (u 0 , u 1 ) ∈ H , for all T > 0 there exists a unique generalized solution u ∈ C(0, T ; H ) depending continuously on the initial data. This implies that the map (u(0), u t (0)) → (u(t), u t (t)) defines a continuous (nonlinear) semigroup S(t) on H . Additionally, the following energy equality holds: Remark 1. Assumption 1 provides linear bounds for the damping function from below and above which ensures the validity of the energy equality (for more details, see [18,Section 4.2]).

Main results and discussion.
3.1. Statement of main results. Our first main result establishes the existence of the compact global attractor for the dynamical system (S(t), H ). As a first step, we show that the semigroup generated by this system is dissipative. This is to say that there exists a bounded absorbing set which captures (in finite time) all trajectories emanating from a given, bounded set of initial data.  (1) taken with (CHD) boundary conditions. This is to say: for all R 0 > 0 and initial data (u 0 , The asymptotic smoothness property of (S(t), H ) is the next supporting result. To achieve this, as in control problems when only a portion of the boundary is subjected to a feedback, we need an additional (standard) star-complemented geometric condition on the undamped portion of the boundary [18,30]: Remark 2 (Geometric Condition and Trace Estimates on Γ 0 ). To provide some context, we note that early control and stabilization studies for hyperbolic-like problems required geometric conditions on the entire boundary (the active and inactive portions) [30] (and references therein). The reference [32] concerning the linear Euler-Bernoulli plate equation with given RHS (which we critically use in [24] and here) allowed the removal of "unnatural" geometric restrictions on the controlled portion of the boundary. (Note that in [24], the main result appeals to the starshaped assumption on Γ 1 = Γ due to the contribution of the nonlinear term on the boundary.) Most control and stabilization considerations (for instance, those in [15,17,18]) retain a geometric assumption on the uncontrolled portion of the boundary Γ 0 -namely the star-complemented condition-for their principal results; this is to provide control of ∆u on Γ 0 , which is not directly controlled by the estimates in [32]. In our analysis, to obtain attractors with a "split" boundary, we require control of the term ∆u Γ0 in the estimates. In utilizing the multiplier approach in [2,3], we will obtain control of this term using the nonlinear structure of the problem in showing dissipativity of the dynamics. This does not eliminate the need for a geometric (sign control) assumption on Γ 0 in showing the asymptotic smoothness property of the dynamics. We will elaborate on this below (see Remark 5).

Remark 3.
When dealing with a solution to (1) we use the nonlinear energy to our advantage. By utilizing the so called trace-moment inequality (Theorem 5.1), and exploiting the super-quadratic nature of the nonlinear energy, we dispense with the star-shaped geometric condition on Γ 1 , which would be analogous to what is assumed in [24].

3.2.
Main results in relation to previous literature. The treatment [24] is the precursor to the work at hand. In [24], the focus is on the general qualitative effects of boundary damping in Berger's (non-rotational) plate model. (In [24], as part of the discussion of earlier results, long-time behavior of Berger plates and beams with interior dissipation is also addressed. 2 ) Thus, the work [24] discusses plate modeling considerations, as well as provides a comparative study between the effect of boundary damping on Berger's versus von Karman's plate for both hinged and free boundary conditions with dissipation. In many configurations (since Berger's plate model is a simplification of von Karman's) the analysis of Berger's plate is subsumed by the analysis of von Karman's plate. In the case of boundary damping, however, this is not so; indeed, for hinged dissipation type boundary conditions and free-clamped dissipation we notice distinct differences in the qualitative behavior of solutions, as well as distinctions in the applicability of certain analytical techniques.
One of the principal results in [24] concerns the existence of a compact global attractor when hinged boundary damping is active in Berger's plate model on the entire boundary. This result utilizes a different technique than what has been used in the analogous configuration for von Karman's plate; specifically, in [24] the absorbing ball is constructed directly-rather than by appealing to a gradient structure for the dynamics (which requires some additional interior damping, as in [17,18]). In [24], a geometric assumption is made on the (controlled) boundary; this assumption was made to accommodate a boundary contribution from Berger's nonlinearity which does not occur in the presence of the von Karman nonlinearity.
In [18], when the (von Karman) dynamics are under hinged boundary dissipation, the damping need not be active on the entire boundary in order to obtain a compact global attractor. Indeed, with a geometric assumption on the uncontrolled, clamped boundary portion (which is also assumed here), active damping via a hinged type dissipation on a strict subset of the boundary will yield the existence of a smooth global attractor of finite dimension. Subsequently, it is a natural question to investigate whether active damping on a portion of the boundary will lead to a compact global attractor for Berger's dynamics. The answer to this question is non-trivial, and the main subject at hand.
In the course of establishing dissipativity-Theorem 3.1-we eliminate the need for a geometric condition (as was appealed to in [24]) on the controlled portion of the boundary. We achieve a smooth global attractor of finite fractal dimension with damping active only on this portion of the boundary. For the existence of the global attractor, our technique is a refinement of the approach in [24], as we must utilize technical trace estimates on the term ∆u| Γ0 (a high order trace on the uncontrolled portion of the boundary) in constructing the absorbing ball. Doing so requires a particular multiplier analysis, and a few critical amendments to the approach in [24] to obtain the compact global attractor. Additionally, we show that dissipativity of the dynamical system can be obtained without any geometric assumptions on Γ.
After showing the existence of a compact global attractor for these (CHD) boundary conditions, we proceed to show via a stabilizability approach the finite dimensionality and additional smoothness of the global attractor constructed at the previous step. This represents a substantial improvement of the results in [24], and brings the analysis of Berger's non-rotational plate with (CHD) boundary conditions inline with the results in [18] for von Karman plate dynamics. (Of course the results on smoothness and finite dimensionality here also apply to the case of an entirely damped boundary-the model in [24].) Lastly, we mention the new treatise [11] on quasi-stable dynamical systems. The abstract analysis in this reference on certain dynamical systems generated by second order evolutions is pertinent to our long-time behavior studies here. See Remark 7 for further discussion.
Owing to the approach described above (due to the special structure of Berger's nonlinearity) our results here represent an improvement over those presented in [18] for von Karman's dynamics with (CHD) in the following senses: 1. We obtain a bound on the size of the global attractor, via a direct dissipativity analysis on the dynamics with no geometric assumptions on the domain. In the most up-to-date analysis of von Karman's non-rotational plate with boundary damping, an indirect approach is used (appealing to a gradient structure for the dynamics); we opt for a direct approach which applies Theorem 5.3. 2. We do not require interior damping in addition to the boundary damping to obtain smooth, finite dimensional global attractors. In [18], this additional damping is necessary and yields the gradient structure for the dynamics, as mentioned in point (1) [14] for the semilinear wave equation. This approach was adapted to our current situation in the proof of [24,Theorem 4.1]. However, at this point we note that unlike [24]-since the boundary has two disjoint portions-the proof has critical differences. It is divided into several steps, which are presented below.
Step 1: Observability estimate We begin with the following estimate:

Lemma 4.1. Any generalized solution u to (1) taken with (CHD) boundary conditions satisfies the following inequality
where the constant C does not depend on T .
Proof. The proof uses the equipartition and flux multipliers u and h·∇u (for . Then, using (5)  Now, in order to estimate the RHS of the above inequality-which involves higher order trace terms-the steps mirror those in the stability analysis of the Kirchoff plate in [28] (which itself critically relies on Theorem 4.3 below-from [32]). However, we approach higher order trace term involving ∂ ν (∆u) in a different way which, ultimately, benefits our analysis and fundamentally exploits the structure of the Berger nonlinearity.
Step 2: Estimating nonlinear terms via higher order traces The steps below are reminiscent of [28], adapted in a few critical places. Let the operator A = ∆, acting on L 2 (Ω) with domain H 2 (Ω) ∩ H 1 0 (Ω), and let D be the associated Dirichlet "lift" map defined by Accounting for boundary conditions, the operator representation of (1) is then Applying A −1 to (10) (justified on strong solutions, and a posteriori on generalized solutions via the estimate produced) we obtain Now, taking the normal derivative of both sides of above equality, multiplying by h · ∇u, integrating over the space-time cylinder [0, T ] × Γ, and reading off from the equation we have the relation where we have noted that The following is the critical step which allows our approach to the higher order trace terms to obtain (as our tack is distinct from [28]). This relies specifically on the structure of the Berger nonlinearity. If we note that and again that h · ∇u = (h · ν)∂ ν u on Γ, we have: Remark 4. In [24] the final term in the above identity is handled by way of a geometric assumption on Γ 1 ; namely by assuming that we have chosen h = x − x 0 so that h · ν ≥ 0 for x ∈ Γ 1 . This allows us to discard the nonlinear boundary contribution. Here, we opt to estimate it directly using the trace-moment inequality (Appendix, Theorem 5.1).
The last equality gives Now, we estimate the integrals on the RHS of the above inequality, term by term. Using integration by parts in time we note that From the elliptic regularity theorem (Γ smooth, with Γ 0 and Γ 1 disjoint), for any h ∈ L 2 (Ω) : ||A −1 h|| 2 ≤ C||h|| 0,Ω ; thus we have: Corollary 1, the Hölder-Young inequalities, and (5) yield that Again, since from standard elliptic theory, ∂ ν D ∈ L (L 2 (Γ), H −1 (Γ 1 )), we have We note that although the first derivatives of ∇u on Γ 1 produces double tangential derivatives ||u τ τ || 2 L 2 (Γ1) , we omit this term since u = 0 on Γ 1 and so ∂ τ u = 0 on Γ 1 . Similarly, by elliptic theory, we obtain Finally, we estimate the nonlinear boundary term From Young's inequality and (4) it follows that ) 2 +C(ϵ, T ). (18) Taking into account (15)- (18) in (14), using (5), and applying the Hölder-Young inequalities we arrive at the next preliminary estimate which can be implemented in (9): For the terms in the last line of (9) involving the higher order trace term ∂ ν (h · ∇u), we note the boundary condition ∆u = −g(∂ ν u t ) on Γ 1 , and we have We now combine (19) and (20) in (9) and absorb terms to obtain: Step 3: Trace estimate for (∆u)| Γ0 By way of estimating (21), we will require the following estimate for (∆u)| Γ0 .

Lemma 4.2.
Let u be the solution to (1) under (CHD) boundary conditions. Then (∆u)| Γ0 ∈ L 2 (0, T ; L 2 (Γ 0 )) with the following estimate: where Proof of Lemma 4.2. For this estimate, we will use a multiplier approach which was also used critically in [2,3] (though it was used earlier in other contexts [30,35]). A vector field as in (23) exists when Γ 0 and Γ 1 are disjoint [2,3]. We then consider the multiplier m · ∇u for (1). Though formally this multiplier has the same structure as the h · ∇u multiplier (with h(x) = x − x 0 for some x 0 ) used in obtaining (9), the specification of m(x) in (23) will aid in the derivation of the estimate (22).
With the vector field m as in (23), multiplying (1) by m · ∇u and integrating on the space-time cylinder Ω × [0, T ], we have: For the first term, we have, exactly as derived in [2]-see p. 18 (and as u t Γ = 0), From here it is clear that The two critical terms in the estimate are the biharmonic term and the nonlinear term. We begin with the biharmonic term. Noting that we have clamped boundary conditions on Γ 0 and that m vanishes on Γ 1 , we have the following equality: We utilize the following facts: (i) the relation (due to u Γ = 0) [28, p. 463]: With (23), these yield: Thus with the above, and the hinged boundary conditions on Γ 1 , (27) becomes: Moreover, for the first term on RHS, we have initially, where ∫ Ω (∆u, m 1yy u x + 2m 1y u xy + m 2yy u y + 2m 2y u yy ) Ω dt. (31) Proceeding, we then rewrite first term on right hand side of (30) as Combining (30) and (32), and subsequently applying the Divergence Theorem of Gauss (and also recalling the definition of m in (4.16)), we have now Using (34) in (29), we have now the key identity: We note that clearly Next, we consider the nonlinear term: where the boundary terms are null, owing to the clamped boundary conditions on Γ 0 and m ≡ 0 on Γ 1 . Explicitly computing ∇(m · ∇u), we have the identity (where H(u) is the Hessian of u, and D(m) the Jacobian of m):
Step 4: Estimating higher order traces via [32] Now, considering (21) over the interval (α, T − α) instead of (0, T ) (hence, performing the calculations above on (α, T − α)), we can use the decreasing nature (modulo a constant) of the energy functional ( (6) and (8)), and then extend some benign integrals back on (0, T ). We obtain: Now, for the second order trace terms ∂ νν u| Γ1 and ∂ τ ν u| Γ1 above, we will use the following sharp regularity result for the boundary traces of solutions to the Euler-Bernoulli equation (linear, with given RHS): Hereafter, the explicit dependence of the above constants on α, δ, γ, h will be suppressed. We now proceed to estimate the key terms in the first line of (39).

G. AVALOS, P. G. GEREDELI AND J. T. WEBSTER
If we take into account the last two inequalities we have: (IV) Returning to (39) and collecting (42)-(44), we have: ] . (45) Now, using the decreasing property of the energy and (6) we finally get Using the properties of the damping (Assumption 1), we note these facts from [18,24]: From which it is straight-forward to obtain (see [24]): where Now, if we take into account (47) in our observability estimate (46) we have Again, via (6) and (8) where C here does not depend on T . Thus, scaling T to be sufficiently large, we have Step 5: Stabilization-type argument Now, if we rewrite the last inequality in terms of the full nonlinear energy E (T ), use the relation (6) between E and E, and employ the notation where M is the constant coming from (7), we obtain: where K(T ) does not depend on E (0) and T > max{1, 2α + C} is large enough but fixed. Using the fact that Note that we can freely specify C(T ) > 1 for T chosen large enough. Rearranging, we have:
Since C(T ) > 1 for the chosen T , we get Now, we can reiterate the same estimate on each subinterval (mT, (m + 1)T ) via the semigroup property. We note that the constants η( E(0), T ) and K(T ) will be same at each step. Then we obtain As is standard, the monotonicity and continuity of the energy functional E , and the fact that η < 1, yield which, taken together with (6), completes the proof of Theorem 3.1.

Proof of Theorem 3.2-Asymptotic
Smoothness. Now, as the second step of the existence of the global attractor, we give the asymptotically smoothness criterion for the solutions to (1). For this, we are interested in the difference of two solutions z = u − w, where U (t) = (u(t), u t (t)) = S(t)y 1 and W (t) = (w(t), w t (t)) = S(t)y 2 solve (1) corresponding to initial conditions y 1 = (u 0 , u 1 ) and y 2 = (w 0 , w 1 ) (respectively), taken from an invariant, bounded set. Then, z will solve the following problem: The energy of the system is given by The relation (6) between the energies gives that there exists an R * such that the set (6) and (8), any bounded set B ⊂ H is contained in W R for some R, and the set W R is invariant with respect to S(t). Then, we consider the restriction of the dynamical system (S(·), H ) to (S(·), W R ) in showing the asymptotic smoothness property, and thus we consider the solutions u, w satisfying The proof of asymptotic smoothness of the dynamical system (S(t), H ) follows the same tack as in [24]. We provide some details here. (3.2). As in the proof of [24,Theorem 4.4], the key point is the observability inequality given in [18, (10.5.15)

Proof of Theorem
where The above inequality is proved in [18,Section 10.3] and is valid for both von Karman and Berger dynamics in the configuration of interest: (1) with (CHD) conditions. It relies on the equipartition and flux multiplier analysis on (49), and critically uses: (i) the sharp trace estimates from (4.3) for the linear plate equation, as well as (ii) the geometric condition on Γ 0 there.

Remark 5.
In the proof of the above inequality we discarded the term via the geometric condition that h · ν ≤ 0 on Γ 0 . We note that in the construction of the absorbing ball the analogous term (coming from the biharmonic term) was accommodated by utilizing ∫ T 0 ( E) 2 dτ , and we dispensed there with the need for the geometric condition on Γ 0 . Now, by the energy relation Combining (i) and (ii), we obtain that there exists T 0 > 0, and constants C 1 (T ) and C 2 (R, T ), such that For detailed calculations see [18, p. 619]. In particular, we are using the fact that the cutoff function β may be chosen so that dt is independent of T > 0. Using the assumption on the structure of the damping (Assumption 1), we have On the other hand, it follows immediately from the energy relation (51) that Now, invoking the decomposition of (F(z), z t ) [24,Theorem 4.6] we have As the analogous bounds on F * and F * * follow immediately-since f (·) is locally Lipschitz-we have now for T ≥ T 0 > 1, Applying the same procedure followed in the proof of [ In the proof, a critical role is played by the following stabilizability estimate valid for the system generated by the solutions to (1).

Lemma 4.5.
Let Assumptions 1 and 2 (for all s ∈ R, as discussed above) be in force. In addition, assume that S(t)y 1 = (u(t), u t (t)) and S(t)y 2 = (w(t), w t (t)) solve (1) corresponding to initial conditions y 1 = (u 0 , u 1 ) and y 2 = (w 0 , w 1 ) (respectively), taken from the set W R . Then there exists positive constants ω, C 1 , and C 2 , C 3 (both depending on R), such that and where z = u − w.
Proof of Lemma 4.5. Our beginning point will be the observability estimate (52). Since by the linear growth condition g ′ (s) ≥ m 1 for every s ∈ R, we have Considering the last inequality in (52) we have which together with (51) and (53) yields where σ = σ(T ) = C1(T ) 1+C1(T ) < 1. Then, the last inequality gives After iteration, we observe that Since σ < 1, we obtain that there exists ω > 0 such that for all t ≥ 0, which gives (55).
Then the Intermediate Derivatives Theorem-see, e.g., Theorem 1.1 of [34]-implies that which together with the application of Gronwall's inequality gives the following a priori bound: Finally, the relation (51) and additional assumption on g yield (56). Now, to finish the proof of (a) in Theorem 3.4 we also need the next assertion; Lemma 4.6. The difference of generalized solutions u − w = z (satisfying (49)) has the property that: where R denotes dependence of the estimate on the ball from which u, w are chosen.
Estimating RHS via (56) and the locally Lipschitz property of the Berger nonlinearity gives the desired result.

Completion of the Proof of (a) of Theorem 3.4
In order to prove the finiteness of fractal dimension of the attractor we utilize Theorem 5.4. For this, we apply the method of "short trajectories", inspired by [36,40] (and used often in [11,18]) and make use of Lemma 4.5 and 4.6. Let us introduce the extended space Then the norm in X is given by . Let A be the global attractor for the semiflow S(t). Consider the set A T ⊂ X which is a suitable extension of A, Our aim is to show that A T has finite fractal dimension in the extended space X. Because, in this case, since the operator P : X → H ; (u 0 , u 1 , z(t)) → (u 0 , u 1 ) is Lipschitz continuous with PA T = A this yields that dim H f A ≤ dim X f A T < ∞, which then concludes the proof.
To this end, on the set A T we define the operator V : A T → X; (u(0), u t (0), u(t)) → (u(T ), u t (T ), u(t + T )) which is a translation by T of the solution. Now, to apply the abstract Theorem 5.4, we need to show that the map V satisfies the conditions of this theorem. The conditions (i) and (ii) in Theorem 5.4 follow easily from the definition of the map V with the invariance property of the attractor A, (56) and Lemma 4.6. For the proof of (iii), we rely on the stabilizability estimate given in Lemma 4.5 and Lemma 4.6.
Invoking (55) twice-and in particular, integrating (59) from T to 2T with respect to t we obtain We mention that the new monograph [11] contains a general definition of quasi-stable dynamical systems (which is broader than the definition in [18]). In essence, a quasi-stable dynamical system is one where the difference of two trajectories can be decomposed into a uniformly stable part and compact part. The theory of quasi-stable dynamical systems has been developed rather thoroughly in recent years [11,18]. In the arguments preceding this section, we opted for a direct proof of finite dimensionality and smoothness via the so called stabilizability estimate. However, with notions of quasi-stability in [11], it seems possible to show that the dynamical system (S(t), H ) above is quasi-stable on the attractor A. At which point, abstract theorems following the quasi-stability property lead to finite fractal dimension and smoothness of the attractor (as well as existence of an exponential attractor). 3

Appendix.
5.1. Trace estimate. The following theorem we use is referred to as the Trace-Moment Inequality [8].
In practice, we utilize it as follows: Corollary 1. Suppose u ∈ H 2 (Ω). Then for ∂ ν u = (∇u) · ν (valid in a collar of the boundary) we have We recall the following useful criterion (first appearing [29] and stated in the present version in [18]) for the asymptotic smoothness: In the theory of infinite-dimensional dynamical systems, finite fractal dimensionality is proved by an approach which is related to the squeezing property. This useful tool is given by the following theorem: iii) There exists compact seminorms n 1 (x) and n 2 (x) on H such that for any v 1 , v 2 ∈ M, where 0 < η < 1 and K > 0 are constants. Here, a seminorm n(x) on H is said to be compact iff n(x m ) → 0 for any sequence x m ⊂ H such that x m → 0 weakly in H. Then M is a compact set in H of finite fractal dimension.