J Dyn Diff Equat (2009) 21:269–314 DOI 10.1007/s10884-009-9132-y

Global Attractor for a Wave Equation with Nonlinear Localized Boundary Damping and a Source Term of Critical Exponent Igor Chueshov · Irena Lasiecka · Daniel Toundykov

Received: 16 August 2007 / Revised: 7 February 2009 / Published online: 18 March 2009 © Springer Science+Business Media, LLC 2009

Abstract The paper addresses long-term behavior of solutions to a damped wave equation with a critical source term. The dissipative frictional feedback is restricted to a subset of the boundary of the domain. This paper derives inverse observability estimates which extend the results of Chueshov et al. (Disc Contin Dyn Syst 20:459–509, 2008) to systems with boundary dissipation. In particular, we show that a hyperbolic flow under a critical source and geometrically constrained boundary damping approaches a smooth finite-dimensional global attractor. A similar result for subcritical sources was given in Chueshov et al. (Commun Part Diff Eq 29:1847–1876, 2004). However, the criticality of the source term in conjunction with geometrically restricted dissipation constitutes the major new difficulty of the problem. To overcome this issue we develop a special version of Carleman’s estimates and apply them in the context of abstract results on dissipative dynamical systems. In contrast with the localized interior damping (Chueshov et al. Disc Contin Dyn Syst 20:459–509, 2008), the analysis of a boundary feedback requires a more careful treatment of the trace terms and special tangential estimates based on microlocal analysis. Keywords Wave equation · Localized boundary damping · Critical source · Attractor · Fractal dimension · Regularity of the attractor Mathematics Subject Classifications (2000)

35B41 · 35L05 · 35B40 · 35B33

I. Chueshov Department of Mathematics and Mechanics, Kharkov National University, Kharkov 61077, Ukraine e-mail: [email protected] I. Lasiecka (B) Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA e-mail: [email protected] D. Toundykov Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA e-mail: [email protected]

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1 Introduction The paper studies long-term behavior of hyperbolic flows generated by a semilinear wave equation with nonlinear boundary damping and a nonlinear source of the critical Sobolev exponent. The goal is to investigate the existence and various properties of global attractors, in particular: their dimensionality and smoothness. While questions of this sort have attracted considerable attention in recent years, the vast majority of existing results pertain to dissipation supported over the entire domain, and either sources of sub-critical growth, or certain special critical settings (e.g. linear damping [2], or damping with a large scaling parameter [9,17,24,26], or polynomially structured nonlinearities [19,40]). Geometrically constrained dissipation, such as one concentrated on a portion of the boundary, results in dynamics where geometric aspects of wave propagation play a crucial role [6]. On the other hand, the criticality of the source associated with the loss of compactness leads to complications at the topological level. In view of these challenges the scarcity of results on attractors for models of this type is hardly surprising. In fact, asymptotic finitedimensionality or smoothness of hyperbolic flows, where the instability is intrinsically an infinite-dimensional phenomenon, should not be necessarily expected. Analysis of attractors for the wave equation with nonlinear damping and critical sources has been an open and recognizably difficult problem even under full interior dissipation. Let us elaborate on the associated challenges. The key role in the analysis of asymptotic regimes of the wave dynamics is played by: (i) the basic energy (in)equality, and (ii) the equi-partition relation—both types of estimates must be established for differences of two solutions. The equipartition equation roughly states that the information encoded in the kinetic energy (related to the damping) can be transferred onto the potential energy, hence providing a gauge on the total energy of the system. In the presence of destabilizing sources, the basic energy identity acquires additional integrals that involve products of the sources with velocities; the criticality of the sources in wave equations results in products which are not compact perturbations of the dynamics. To better illustrate this aspect consider the three major modes of nonlinear dissipation: (i) full interior damping, (ii) localized interior damping, and (iii) boundary damping. (i) Full-interior dissipation. In this case the damping provides complete information on the kinetic energy. Thus, the equipartition equation can be attained by means of “lower order” multipliers which contribute to compact perturbations. The only difficulty to be resolved is the handling of the critical term in the energy relation. However, the latter quantity is typically controlled by the dissipation integral—owing to the fact that the damping is supported on the entire domain. This technique was successfully implemented in [10–12]. (ii) Localized interior damping. This is a much more subtle scenario since the dissipation is geometrically constrained to an open set in
. The reconstruction of kinetic energy now requires weighted multipliers that “propagate” the energy from the controlled area onto the entire domain. These “flux multipliers” are of the energy order, hence are no longer compact with respect to the topology of the state space. The resulting integrals will contain products of the source with the multiplier terms, thus contributing to the loss of compactness. The technique developed recently in [15] and based on Carleman’s estimates has proved successful in handling this case. However, the fact that the damping is supported on an open set plays a critical role in the arguments of [15]. Another key difficulty is that the basic energy identity contains, as mentioned before, integrals which involve velocities of the solution not supported within the dissipative

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area, i.e. disjoint from the control of the kinetic energy offered by the damping. For this reason the bound on the dissipation cannot help fully estimate the critical products of the source with the velocity, and the aforementioned method of “dissipation integrals” used for the interior damping is no longer applicable. (iii) Boundary damping. In addition to the complications of the localized interior case, models with boundary damping pose two more difficulties. First, the “flux” multipliers in the Carleman estimates cannot be compactly supported in
because the damping must be “propagated” from the boundary rather than an open set. As a result, new tangential boundary derivatives, uncontrolled by the dissipation, enter the estimates; these boundary terms are super-critical, as they require topologies strictly above the finite energy level. Second, in the interior/localized interior cases the damping provided at least partial (e.g. on an open subset) control of the velocity in the interior; but the boundary dissipation, being concentrated on a set of measure zero, offers no direct means to assess the velocity inside the domain, not even on a thin a boundary collar. The main contribution of this paper is the proof of smoothness and finite fractal-dimensionality of attractors for the dynamics governed by a critical wave equation with nonlinear boundary damping—case (iii) above. In addition to the issues raised therein, only partial boundary observation is assumed which, along with the failure of the Lopatinski condition, prevents applicability of the now-standard radial flux multipliers. Different classes of multipliers, accounting for this geometric scenario have to be constructed analogously to [15,32]. New technical challenges stem from the appearance of the aforementioned super-critical boundary traces in the calculations. These terms are handled by microlocal tangential boundary estimates developed originally in the context of stabilization of wave equations with Neumann conditions. Summarizing: the methods employed earlier in the literature are inadequate to deal with a critically perturbed wave equation with nonlinear damping constrained to the boundary (a portion thereof). These features introduce novel challenges at the level of handling criticality in: (1) kinetic energy, (2) potential energy, and (3) boundary terms; thus, calling for a new set of mathematical tools, which are presented in the proofs below. 1.1 The Model Let
⊂ R3 be a smooth bounded connected domain with boundary . We will focus on a 3-dimensional model since it captures all the difficulties that arise in higher dimensions and easily simplifies to lower-dimensional settings. Let Q T :=
×]0, T [, T := ×]0, T [. Consider the following semilinear wave equation: wtt − w = f (w) w(0) = w0 , wt (0) = w1

in Q T in


(1)

with the boundary condition ∂w (2) + w + χ(x)g(wt ) = 0 on T ∂ν The initial data {w0 , w1 } belongs to a suitable function space, to be specified later; maps g(s) and f (s) are real-valued functions on R; the Nemytski operators wt → g(wt ) and w → f (w) model respectively the frictional damping and the source term. The function g(s) is continuous and monotone increasing. The nonnegative cutoff map χ restricts the action of g to a subset of the boundary: χ ∈ C 1 (, [0, ∞[)

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We assume there exists a nonempty segment of the boundary C (for “Controlled”) where χ has a positive bound from below: inf χ(x) > 0

x∈C

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Remark 1 (Regularity of the cutoff) To guarantee the existence of strong solutions it suffices to let χ be a multiplier on H 1/2 (), e.g. χ ∈ L ∞ () ∩ W 1/2, 4 () (see [39, Sect. 2.3.1]); however, to carry out the inverse energy estimates we will need χ to act as a smooth cutoff. The remaining portion of the boundary (which, unless empty, will actually overlap with C ) is denoted geom (for Geometry):  = C ∪ geom

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The segment C is assumed to be connected, relatively open, and have a positive measure. When non-empty, the set geom should possess the same properties and, in addition, must satisfy certain convexity-type assumptions (specified later in Sect. 1.3), which are necessary for the restricted damping to be effective under Neumann-type boundary conditions. Note that since C and geom form an open cover of the connected boundary , we implicitly assume that the two portions overlap (which will be used later on). Also observe that due to the boundary data (2) the system (1) does not satisfy the Lopatinskii condition. 1.2 Assumptions on the Nonlinear Terms In order to study the long-term behavior we must first guarantee global existence of solutions. For that purpose some additional assumptions must be imposed on the nonlinear functions f, g: Assumption 1 (The source) The source term is modeled by f ∈ C 2 (R) satisfying 1. In the 3-dimensional case, let the map f be of at most critical (cubic) order, which follows from: | f  (s)| ≤ C f (1 + |s|)

(5)

lim sup f (s)/s < λ0

(6)

2. Also assume that |s|→∞

where λ0 > 0 is the minimal eigenvalue of the elliptic spectral problem
−u = λu in
∂u + u = 0 on  ∂ν Denote the corresponding negative Robin Laplacian by A := −. Due to the condition (5) and basic Sobolev embedding results, the function f corresponds to a locally Lipschitz Nemytski operator H 1 (
) → L 2 (
). In particular, the following result is standard: Proposition 1 (Locally lipschitz source) f (w) − f (u) L 2 (
) ≤ Cr w − u H 1 (
) ∀ w H 1 (
) , u H 1 (
) ≤ r < ∞

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Remark 2 When dim
= 2, the assumption on the growth of the source may be replaced by any polynomial growth (since the continuous embedding H 1 (
) → L p (
) holds for 2 ≤ p < ∞ in 2D). The two-dimensional case is easier to analyze and will not be dealt with separately. Higher-dimensional domains
⊂ Rn>3 can be treated similarly by imposing the appropriate (critical) growth condition on the map f . We will concentrate on the 3D framework since it captures all the difficulties related to the study of the critical exponents. Assumption 2 (The damping) Let g ∈ C 1 (R) be monotone increasing with g(0) = 0. Assume there exist positive constants m and M such that m < g  (s) < M ∀s ∈ R

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Remark 3 (Existence of the attractor) When verifying the existence of the global compact attractor, Assumption 2 can be relaxed by requiring only g ∈ C(R) and asserting that for any η > 0 there exists a constant Cη > 0 such that  2   g(s1 ) − g(s2 ) + (s1 − s2 )2 ≤ η + Cη g(s1 ) − g(s2 ) (s1 − s2 ) (9) for all s1 , s2 ∈ R. 1.2.1 Comments on the Assumptions 1 and 2 The order of the source f (w) can reach the critical level (O (w 3 ) for
⊂ R3 ) allowed by the well-posedness theory: it corresponds to the largest admissible exponent (in the polynomial bound on f ) for which f maps H 1 (
) into L 2 (
). Thus, the restriction of the nonlinearity up to the critical level is a natural one when studying long-term behavior, and sources of higher order rise the issue of well-posedness of the solutions in the space of finite energy (∼ H 1 (
) for displacement/acoustic velocity potential w; and L 2 (
) for the velocity/pressure wt ), and may lead to a blow up of solutions in finite time [21,42,44]. Due to the restricted support of the dissipation, linear bounds at infinity implied by (8) (or (9)) are necessary not only for the existence of a global attractor, but even for uniform decay results to a single equilibrium. Counterexamples are known in some 1-dimensional cases: e.g. [48, Theorem 4.1] proves that under a boundary feedback that is linear at the origin and saturated at infinity (g(s) = sgn(s) for large |s|) the asymptotic decay of the energy is not uniform with respect to the initial state, hence there can be no set (with a nonempty interior) that is uniformly attracting with respect to the finite energy topology. For a relation between the energy decay and topological properties of the solutions see also [16,32]. As far as the finite-dimensionality is concerned, the question whether one can relax the condition (8) near the origin still remains open even in the situation of a full interior damping. 1.3 Geometry of the Domain In order for a geometrically restricted damping to be effective it is necessary to impose constraints on the shape of
[6]. In this case the difficulty stems from the “non-dissipative” part of the boundary where the Lopatinskii condition is not satisfied, or more specifically— due to the Neumann-type conditions on the uncontrolled boundary segment (this obstacle does not arise in the Dirichlet case due to the well-known “hidden” regularity [35] for the wave problem). To cope with this issue we follow [36] and construct weight functions for Carleman estimates using special convexity assumption on the uncontrolled region of : Assumption 3 (Geometry for the case of uncontrolled boundary: when geom = ∅) If the uncontrolled segment geom is non-empty, assume that

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(a) geom is a level set geom = {x ∈ Rn : (x) = 0},  ∈ C 4 , ∇ = 0 on geom where map  is defined on a suitable domain in (b) Assume one of the following:

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R3 .

• there exists x0 ∈ Rn so that (x − x0 ) · ν(x) ≤ 0 on geom and the Hessian matrix of  is non-negative definite on geom  H  ≥ 0. geom

• Or, instead, (x − x0 ) · ν(x) ≥ 0 on geom and H is non-positive definite on geom  H  ≤ 0, geom

with ν denoting the outward normal vector field on . It is obvious that the condition (b) in Assumption 3 is valid automatically if the either the domain
is convex near geom or the exterior domain is convex near geom . However these conditions are far from necessary as shown in [36], where many other examples (of not convex/concave domains are presented). The Assumption 3 permits to construct a special smooth vector field on
that will eventually help estimate the traces of the solution on geom : Proposition 2 (Vector field h [36, p. 302]) Suppose Assumption 3 holds. Then there exists a strictly convex scalar function d(x) ∈ C 3 (
) whose gradient is tangent to geom . More precisely: there is ρ > 0 such that Hd ≥ ρ I on


(11)

where Hd denotes the hessian of d. Equivalently, if we define the following vector field on
: h := ∇d

(12)

and let Jh stand for the Jacobian of h, then (11) is equivalent to Jh ≥ ρ I

(13)

h · ν ≡ 0 on geom

(14)

In addition h satisfies

Another important observation is that if geom = ∅ then the connected sets C , geom form an open cover of the compact connected boundary . Consequently the set {x ∈  : h(x) · ν(x) = 0} is compactly contained in C : supp (h · ν) ∩  ⊂⊂ C ⊂ supp χ ⊂ 

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A two-dimensional section of a suitable domain can be viewed in Fig. 1. Remark 4 When studying controllability and stabilization with Dirichlet-type boundary conditions, one may consider other geometric configurations where the controlled part of the boundary may not necessarily be a connected set. This was done in the literature by using so-called rotating multipliers [7]. However, this method does not appear applicable to nonLopatinski problems, as of interest to this work.

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Fig. 1 Example of a 2-dimensional cross-section of
. The damping is restricted by the cutoff χ to a subset of the boundary. The support of χ can be arbitrary as long as it contains any set C where inf χ > 0, and C overlaps with the complementary portion geom , which, in turn, obeys certain conditions dictated by geometric optics

1.4 Basic Notation Henceforth use (·, ·), ,  to denote the inner products in L 2 (
) and L 2 () respectively. With a slight abuse of set-theoretic notation, “{a1 , a2 , . . . , an }” will stand for an ordered n-tuple “(a1 , a2 , . . . , an ).” For convenience, in addition to Q T , T , define: Q ts :=
×]s, t[, st := ×]s, t[. Next, • · s, p,X will indicate the norm in the Sobolev space W s, p (X ). • · s,X will be a shorthand for the norm in H s (X ). • · := · 0,
= · L 2 (
) To analyze global attractors we will study differences of evolution trajectories, hence the following notation will appear frequently: z := w − u

(16)

and slightly abusing the notation: g(z ˜ t ) = g˜ u (z t ) := g(z t + u t ) − g(u t ) = g(wt ) − g(u t ) f˜(z) = f˜u (z) := f (z + u) − f (u) = f (w) − f (u)

(17) (18)

(though g(z ˜ t ), f˜(z) depend on u to uniquely determine w = u + z, the subscript “u” will be understood throughout and omitted to avoid “cluttering” the calculations). Equations 1 and 2 imply that z satisfies z tt − z = f˜(z) z(0) = z 0 := w0 − u 0 , z t (0) = z 1 := w1 − u 1

in Q T in


(19)

and ∂z t + z + χ(x)g(z ˜ t ) = 0 on T ∂ν Reserve a few more symbols for the “dissipative” products: D(wt ) := χg(wt ), wt ,

(20)

˜ t ) := χ g(z D(z ˜ t ), z t 

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1.5 Well-posedness The existence and uniqueness of solutions to system (1) and (2) follows from the theory of monotone operators (for a general reference see [5] or [45]). The evolution from the initial state Y0 = {w0 , w1 } can be represented as a nonlinear semigroup flow S(t)Y0 on the finite energy space H := H 1 (
) × L 2 (
)

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The strong solutions correspond to the initial data from the domain of the appropriate evolution generator, whereas generalized (weak) solutions are defined as strong limits of strong solutions. The following results, obtained via the monotonicity and nonlinear semigroup methods, are known (see [31, Theorem 2.1], [13, Theorem 2.1] and [10, Theorem 1.3]): Theorem 1 (Existence and regularity of solutions) I. (Generalized solutions) Let {w0 , w1 } ∈ H , then there exists a unique generalized solution {w, wt } ∈ C ([0, ∞); H ) to (1)–(2). In addition, for any T < ∞ ∂w ∈ L 2 (T ), χ 1/2 wt ∈ L 2 (T ), ∂ν

∂w + w + χg(wt ) ≡ 0 in L 2 (T ) ∂ν

The generalized solutions satisfy the following variational equality with derivatives understood in the sense of distributions: d (wt , φ) + (∇w, ∇φ) + χg(wt ) + w, φ = ( f (w), φ) ∀φ ∈ H 1 (
) dt II. (Strong solutions) Assume, in addition, that the initial data {w0 , w1 } belongs to H 2 (
)× H 1 (
) and satisfies the compatibility condition ∂ν w0 + w0 + χg(w1 ) = 0 on . Then the corresponding weak solution is strong and possesses the following regularity:   {w, wt , wtt } ∈ L ∞ 0, T ; H 2 (
) × H 1 (
) × L 2 (
) ∀T < ∞ 1.6 Previous Results and the Contribution of this Paper Attractors for damped semilinear wave equations have drawn considerable research efforts in recent decades. The pioneering works in this area [3,22,23,25,46] studied linear damping supported on the entire domain
, and subcritical sources. Handling of sources with critical exponents in wave dynamics has turned out to be a much more subtle issue due to the intrinsic lack of smoothing mechanism provided by the flow which, in turn, results in the loss of compactness. A very elegant approach resolving this problem in the case of a linear damping and a critical source appeared in a relatively recent paper [4]. However, the method employed in [4] critically exploits the linearity of the damping. Nonlinear dissipation introduces further complications stemming from the infinite-dimensional nature of instabilities in free wave dynamics. Many researchers have provided contributions to this topic; however, the assumptions they imposed involved either sub-criticality of a source, structured nonlinearities, or a damping with a large coefficient [9,18,25,26]. “Supercritical” damping and sources with imposed correlation between the two were successfully dealt with in [19]. The “double-critical” case of a wave equation with a critical source and a critical damping (the latter corresponds to the polynomial growth of order 5 in three dimensions [25]), had been an open problem until fairly recently. Ultimately, its solution required new techniques in

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both dynamical systems [12,30,41] and in controllability theory, resulting in special inversetype observability inequalities [10,12]. For a more detailed overview of attractors for waves with nonlinear damping fully supported in
see [12] and references therein. Now we turn to a more difficult question: what happens when the dissipation is geometrically constrained? It is not surprising that the analysis of this situation is much more delicate, since it involves “propagation of the damping” across the domain. In what follows we shall briefly describe the results available for the geometrically constrained case, which is the one of relevance to this paper. Wave equation with nonlinear damping localized to suitable collars of the domain was treated in [40], which established existence of compact attractors for subcritical sources, and in [20] for a critical source. In the case of linear boundary damping and subcritical source the existence of a global attractor was established in [30] (see also [27] for continuity properties of this attractor). Nonlinear boundary damping was first considered in [13] which proved the existence of a global compact attractor with the dissipation concentrated on the entire boundary of the domain; the analysis in [13] includes critical exponents. Finite-dimensionality of the attractor is more difficult to establish, since it involves Lipschitz-type of estimates performed on the difference of two solutions. Certain “cancellations” that may occur for a single solution, may not be valid for the differences, leading to intractable non-compact perturbations in the estimates. This is certainly the case of the wave equation with a nonlinear damping and critical-exponent sources. Finite dimensionality of attractor with boundary dissipation was ascertained in [14], however, for a subcritical source only. Recent paper [15] proved existence and finite-dimensionality of a global compact attractor for a semilinear wave equation with a critical source term and localized interior damping, which affected only a subset of a boundary collar. The key to handling both geometrically constrained damping and the criticality of the source lies in a careful application of some recently developed abstract tools that characterize the Hausdorff dimension of compact sets [9,10,12], along with inverse observability estimates which build upon methods developed in [36]. One of the key ingredients in [15] was the construction of flux-multipliers based on Carleman’s estimates. A critical role in this argument was played by the fact that the damping was supported on an open subset of the domain. The damping restricted to the boundary, as analyzed in this paper, brings in additional difficulties. Besides the fact that the “flux” multipliers lead to non-compact perturbations, hence necessitate Carleman’s estimates, these high-order multipliers cannot be localized to “suitable” subsets of the domain (e.g. near the part of the boundary which satisfies additional geometrical restrictions) since there is no damping in the interior to take care of the rest. The dissipation being concentrated on a set of measure zero prevents the use of cutoff functions supported on open subsets of the domain. This problem leads to the appearance in the estimates of high-order boundary traces, which are likewise not controlled by the dissipation (tangential gradients). However, these trace terms can be assessed by means of microlocal estimates developed in the layer near the boundary. This procedure essentially shows how to absorb tangential boundary derivatives into the damping and interior terms that of a lower-order with respect to the topology of the state space. Thus, loosely-speaking, the energy “spill-over” from the measure-zero set is controlled by the damping on that set and interior terms, which are, however, “compact”. The downside of the method is that the recovered information on the tangential boundary derivatives requires observation over a possibly longer period of time. This issue, while introducing new technical difficulties at the level of long-term behavior, can be handled by an appropriate time-slicing procedure introduced in this paper.

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Thus, the main contribution of this work is the investigation of attractors within the setting of a critical (non-compact) perturbation and a nonlinear damping feedback g(wt ) concentrated on a portion of the boundary. The goal is to prove that asymptotic regimes of such a system are smooth and finite-dimensional. From the technical perspective the novel challenges encountered include: • The effect of the sources on the recovery of potential energy leads to uncontrolled interior integrals of the critical (energy) level. • Due to the influence of the source, the kinetic energy terms are likewise critical. While the same issue was already present in problems with full interior damping, the boundary dissipation offers much less control. It is clear that L ∞ (0, ∞) bound on the boundary velocity does not imply the same estimate for the interior terms, and, in fact, does not even fully take care of the velocity traces since the dissipation affects just a subset of the entire boundary. To handle this problem the proof exploits the structure of the attractor, along with the previously established compactness property of the attracting set. The main idea of this approach is to propagate the near-equilibrium smoothness of trajectories through the attractor from −∞ forward in time. • While the “decoupling” of the potential energy from the sources is achieved via Carleman’s estimate with a large parameter, as in [15], the inequality is now “tainted” by the appearance of boundary terms that are supercritical (above the topology of the state space). This phenomenon is specific to boundary dissipation, because the damping acts on a set of measure 0 and, therefore, high-order propagation multipliers must be applied throughout the entire domain (in contrast with interior-localized dissipation which permitted the “flux” multipliers to be restricted only to the undamped portion of the interior). To cope with the resulting space-time tangential boundary traces, special estimates based on microlocal analysis are developed. Remark 5 The approach presented in this paper applies to other hyperbolic flows with localized damping, for example Kirchhoff plates accounting for rotational inertia (Euler-Bernoulli plate equations are simpler due to the infinite speed of signal propagation which eliminates the need for the Carleman estimates).

2 Main Result 2.1 Basic Definitions Definition 1 (Global attractor (e.g. see [3,8,46])) A bounded closed set A ⊂ H is said to be a global attractor of the dynamical system {H , S(t)} if (i) A is strictly invariant: S(t)A = A for t ≥ 0 (ii) A is uniformly attracting: for any bounded set B ⊂ H lim dist H (S(t)B , A ) = 0

t→+∞

To describe the attractor, we introduce the set N of stationary points of the flow S(t): N := {W ∈ H : S(t)W = W ∀t ≥ 0}

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Every stationary point W ∈ N has the form W = (w, 0) and solves the elliptic boundary value problem    ∂w − w = f (w) in
; + w  ≡ 0 (23) ∂ν  Definition 2 (Unstable manifold of stationary points) Suppose a curve in the  there exists  phase-space t  → Y (t) ∈ H , t ∈ R, such that the segment Y [a, ∞[ coincides with the evolution trajectory {S(t)Y (a) : t ≥ 0} for all a ∈ R. We will call such Y (t) a full trajectory. Define the unstable manifold M u (N ) emanating from N as the set of all full trajectories Y (t) for which lim dist H (Y (t), N ) = 0

t→−∞

Definition 3 (Fractal dimension, e.g. see [8]) The fractal dimension dim f M of a compact set M can be defined by the formula dim f M := lim sup r →0+

ln N (M , r ) ln(1/r )

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with N (M , r ) being the minimal number of closed sets of diameter 2r which cover M . 2.2 Existence, Fractal Dimension, and Regularity of the Global Attractor The main result of this paper is: Theorem 2 Suppose Assumptions 1 and 2 hold. In addition, if the uncontrolled segment geom is nonempty let Assumption 3 on the geometry be satisfied as well. I. The dynamical system {H , S(t)} possesses a global compact attractor A ⊂ H that coincides with the unstable manifold of stationary points: (i) A = M u (N ) (ii) lim dist H (S(t)W, N ) = 0 for any W ∈ H . t→+∞

II. The fractal dimension of the global attractor A is finite. III. Furthermore, A is bounded in H 2 (
) × H 1 (
). Remark 6 The Part 2 of Theorem 2 holds under a slightly weaker assumption on the damping: one could replace the inequality (8) of Assumption 2 with (9). Remark 7 Existence of compact attractors for the system (1) and (2) with a damping supported on the entire boundary (χ(x) ≡ 1) was proved in [13]. The statement in Part 2 of Theorem 2 now permits to consider damping with restricted boundary support. The technical difficulty of this new setup is the failure of the Lopatinskii condition; as a consequence, one requires very special weight functions for the inverse estimates. In addition, the proof we present is very different from the argument in [13], which heavily relied on the multiplier theory for Besov spaces [39] and the associated smoothness of the damping. What we consider as the most important contribution of this work are the Parts 2, 2 of Theorem 2. To the authors’ best knowledge, this is the very first result claiming finitedimensionality and smoothness of attractors for wave flows with localized nonlinear boundary damping and a critical source.

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3 Proof of the Main Theorem 3.1 Overview 3.1.1 Basic Facts on Attractors The proof relies on new criteria developed in [12] (see also [9–11]) for studying finite-dimensionality and regularity of attractors for abstract 2nd-order evolution equations. Below, for the reader’s convenience, we state these criteria in the form that is convenient for applications to our model. We start with the following classical definition (see, e.g., one of the sources [3,8,23,43, 46]). Definition 4 (Lyapunov function) Let {X , S(t)} be a dynamical system with the phase space

X and evolution semigroup S(t).

• The continuous functional (Y ) defined on X is said to be the Lyapunov function for the dynamical system {X , S(t)} if and only if the function t  → (S(t)Y ) is a non-increasing function for any Y ∈ X . • The Lyapunov function (Y ) is said to be strict if and only if the equation (S(t)Y ) = (Y ) for all t > 0 and for some Y ∈ X implies that S(t)Y = Y for all t > 0, i.e. Y is a stationary point of {X , S(t)}. • The dynamical system {X , S(t)} is said to be gradient if and only if there exists a strict Lyapunov function on X . It is well-known (see, e.g.,[3,46]) that if a system {X , S(t)} is gradient and possesses a compact global attractor A , then for this system the properties (i) and (ii) stated in Part I of Theorem 2 hold. Below we will see that the system {H , S(t)} on the state space H := H 1 (
) × L 2 (
) generated by (1) and (2) is gradient with a Lyapunov function given by the energy functional (see (31) below). To state a criterium for existence of a compact global attractor we need the notion of asymptotic smoothness (see [23]). Definition 5 (Asymptotic smoothness) A dynamical system {X , S(t)} is said to be asymptotically smooth iff for any bounded set B such that S(t)B ⊂ B for t > 0, there exists a compact set K in the closure B of B , such that lim dX {S(t)B | K } = 0,

t→+∞

where dX {A |B } := supY ∈A distX (Y, B ) is the Hausdorff semidistance. The asymptotic smoothness property guarantees the compactness of bounded trajectories. For other asymptotic compactness properties closely related to this one we also refer to [3,8,43,46]. Now we are in a position to state a criterium for existence of a compact global attractor for gradient systems which we use in our considerations. Theorem 3 Suppose {X , S(t)} is a gradient asymptotically smooth dynamical system. Assume its Lyapunov function (Y ) is bounded from above on any bounded subset of B and the set  R = {Y : (Y ) ≤ R} is bounded for every R. If the set N of stationary points of {X , S(t)} is also bounded in X , then {X , S(t)} possesses a compact global attractor A = Mu (N ).

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This theorem easily follows from [23, Theorem 2.4.6] (see also [43, Theorem 4.6]). For details see the proof of Corollary 2.29 in [12] Other criteria for existence of compact global attractors can be found in [3,8,23,43,46] (see also the references therein). All these criteria available in the literature require some ultimate boundedness and compactness properties of trajectories emanating from bounded sets. To establish asymptotic smoothness of the system generated by (1) and (2) use the following assertion which relies on the idea presented in [28]. Theorem 4 (Criterion for asymptotic smoothness) Assume that for any bounded positively invariant set B in a state space X and for any ε > 0 there exists T ≡ T (ε, B ) such that dX (S(T )Y1 , S(T )Y2 ) ≤  + ε,B,T (Y1 , Y2 ), ∀Yi ∈ B ,

(25)

where ε,B,T (Y1 , Y2 ) is a function defined on B × B such that lim inf lim inf ε,B,T (Yn , Ym ) = 0 m→∞

n→∞

(26)

for every sequence {Yn } ⊂ B . Then {X , S(t)} is an asymptotically smooth dynamical system. We note that the present version of the compactness criterion provides more flexibility by allowing taking sequential limits (in n and m) rather than the simultaneous limits. This was an observation made for the first time in [28] for von Karman equations. The result stated in Theorem 4 is an abstract version of Theorem 2 in [28]. For a short proof we refer to the argument given in [11, Appendix A] or in [12, Proposition 2.10]. To prove finite dimension of the attractor we use the following assertion which is a version of criterion given in [9, Theorem 3.11] (see also [11, Theorem 2.2], [12, Sect. 2.2], and [15, pp. 476–477]). We state it in the form convenient for the application below. Theorem 5 Let {H , S(t)} be the system generated by (1) and (2) in the space H = H 1 (
)× L 2 (
). Assume that it possesses the global attractor A and there exist non-negative scalar functions a(t), b(t) and c(t) on R+ such that (i) a(t) and c(t) are locally bounded on [0, ∞[, (ii) b(t) ∈ L 1 (R+ ) and limt→∞ b(t) = 0 (iii) And for every Y1 , Y2 ∈ A and t > 0 the following relations hold S(t)Y1 − S(t)Y2 2H ≤ a(t) · Y1 − Y2 2H

(27)

and S(t)Y1 − S(t)Y2 2H ≤ b(t) · Y1 − Y2 2H + c(t) · sup w 1 (s) − w 2 (s) 2 (28) 0≤s≤t

Here we denote S(t)Yi = {wi (t), wti (t)}, i = 1, 2, and {u, v} 2H = u 21,
+ v 20,
. Then the attractor A has a finite fractal dimension. We note that idea of the proof of this theorem uses a space of “pieces” of trajectories, and goes back to the method of the so-called short trajectories suggested in the paper [37] and developed in [38]. See more details and comments in [12, Sect. 2.2].

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3.1.2 Main Line of Argument To apply Theorems 4 and 5 in our situation one must establish certain inverse-observability inequalities. Let us begin by formally reviewing the steps and main technical ingredients of the general argument (e.g. see [12]) that establishes existence, compactness, and a bound on the fractal dimension of attractors: • If any two evolution trajectories converge to each other (uniformly with respect to the initial energy) up to a perturbation that exhibits certain compensated compactness (with respect to the phase space topology) then we can guarantee the hypotheses of Theorem 4. The Parts II and III of Lemma 1 essentially restate the abstract Theorem 4 in the context of our wave equation. In addition, if the dynamical system is of a gradient type with a “good” Lyapunov function and its set of equilibria is bounded, then by the Theorem 3 the asymptotic smoothness property is equivalent to existence of a global compact attractor that coincides with the unstable manifold of stationary points. • By Theorem 5, if a stronger result holds: the evolution trajectories converge to one another up to a perturbation by seminorms which are compact with respect to the finite energy level, then the attractor has finite fractal dimension. This statement is equivalent to the estimate given in Part III of Lemma 2 below. In general, asymptotic behavior of trajectories can be inferred from the dissipative laws and from the energy relations obeyed by the system, where by the energy at time t we usually mean square of the state-norm at t. Roughly speaking, a basic dissipative law typically states: T [energy at T ] +

[damping on supp χ] = [initial energy] + [source perturbation] 0

An observability estimate would present an additional reconstruction of the energy at some time T (dependent on the speed of signal propagation) via the dissipation:   T  [damping on supp χ] + [perturbation] [energy at T ] ≤ C T h

(29)

0

where C T depends only on T and possibly on the norm of the initial state; and h(s) is a suitable concave increasing function determined by the damping. Note that when the dissipation is restricted (supp χ

), such a reconstruction bears an inverse character since the recovery of the state energy over the entire domain
must be accomplished via the damping which only acts on a subset of
(or a subset of its boundary ). If all the “perturbation” terms are absent (e.g. f = 0, or f satisfies suitable stability conditions), and h(·) is linear, then combining the last two relations with the monotonicity of the damping would show that statenorm at time T is a fixed fraction of the initial energy, whence solutions decay exponentially to 0 (and uniformly with respect to the size of the initial data). As Theorems 4 and 5 indicate, in order to conclude existence and finite-dimensionality of attractors, one needs to derive inequalities of the form (29) for a difference of two solutions. This argument runs into the difficulty arising from the loss of certain “structures” available when carried out for a single solution only. The presence of a critical locally Lipschitz source in general brings in new terms that cannot be absorbed by finite-energy norms and may at best be shown to possess some compactness-like properties sufficient to establish existence

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of attractors. The argument for the finite-dimensionality is yet more challenging, because one must somehow decompose the intrinsically non-compact source-produced terms into “tractable perturbations” of genuinely compact seminorms on the state space. An observability estimate of type (29) valid for a difference of two solutions can be verified as follows: 1. Establish, via Carleman’s estimates, a fundamental identity which provides a decomposition of the kinetic and potential energy components in terms of boundary data, initial conditions, and geometrical features of the domain. The role of the geometry is important since one must verify that, as time goes by, the damping confined to a small subset of the boundary can affect the energy of the entire system. It is at this step when “unwanted” boundary perturbations of the first order must be eliminated by using microlocal estimates. 2. Turn the fundamental identity into an inverse observability estimate which reconstructs the energy of the system via the dissipation, and compact or “compact-like” terms. In the presence of a critical source, the challenge is to actually prove that the non-dissipative terms are benign, in particular: either possess certain compensated compactness, or are almost compact (up to an arbitrarily small fraction of the energy at the expense of other lower-order terms). Thus Theorem 2 is a result of the following steps: (i) Section 3.2.1 defines the energy functionals and proves the gradient property of our dynamical system (Proposition 4). (ii) The bound on the set of equilibria readily follows if we multiply the stationary elliptic problem (23) by w and invoke the estimate (6). Thus the system {H , S(t)} is of a gradient type, with a bounded set of equilibria points, and hence we have the possibility to apply Theorem 3 by checking boundedness properties of the full energy (Corollary 1). (iii) Part I of Lemma 1 and Part I of Lemma 2 establish the fundamental energy identities which are a technical step on the way to desired observability estimates for, respectively, the existence, and the finite-dimensionality results. (iv) Parts II and III of Lemma 1 will provide respectively the observability estimate, and the weak sequential compactness property of the critical perturbation terms. These two results imply the hypotheses of Theorem 4 for the wave system {H , S(t)}, generated by (1) and (2), hence the latter possesses a global compact attractor coinciding with the unstable manifold of stationary points. (v) Part II of Lemma 2 is an intermediate step on the way to observability inequality which implies the hypotheses of Theorem 5 for the system {H , S(t)}, hence implies a (finite) bound on the fractal dimension of the attractor (see also [15] for the case of localized interior damping). With some additional work the regularity of the attractor will follow as well. 3.2 Some Preliminary Definitions and Estimates 3.2.1 Energy Functionals Define the quadratic energy to be the square norm of the corresponding state: E w (t) = E(w(t), wt (t)) := 21 A1/2 w(t) 2 + 21 wt (t) 2 = 21 {w(t), wt (t)} 2H

(30)

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where A = −





D (A) = u ∈ L (
) : u ∈ L (
), 2

2

We can also define full energy as

  ∂u  +u  =0 ∂ν 



Ew (t) = E (w(t), wt (t)) := E w (t) −




F(w(t))d


(31)

where F stands for the antiderivative of the source map: F  (s) = f (s). Proposition 3 (Basic energy identities) Let {w0 , w1 }, {u 0 , u 1 } be smooth initial conditions and w, u denote the corresponding strong solutions to (1) and (2). For any times s, t on the evolution trajectory we have t E w (t) +

t D(wt )dt = E w (s) +

s

( f (w), wt )

(32)

s

or, equivalently, t Ew (t) +

D(wt ) = Ew (s)

(33)

s

Similarly, if z = w − u we have t E z (t) +

˜ t ) = E z (s) + D(z

t

s

( f˜(z), z t )

(34)

s

Proof Equation 32 is the classical energy identity for the wave equation; it follows if we multiply (1) by wt and integrate by parts over the space–time cylinder Q T . Equation 34 can be derived analogously if we instead apply the multiplier z t to Eq. (19). Identity (33) is just a restatement of (32) using the definition (31).   Corollary 1 (Global bounds on trajectories from bounded sets) For monotone continuous damping g (with g(0) = 0)) and under the Assumptions 5 and 6 on the source term f , the energies E w (t), Ew (t) of the solution to (1) and (2), and the dissipation integral remain globally bounded in time: t E w (t) ≤ CB , Ew (t) ≤ CB , and

D(wt (s))ds ≤ CB s

where the constant CB depends only on the diameter of the bounded set B ⊂ H that contains the initial state {w0 , w1 } (i.e. could instead indicate the dependence of C on the initial quadratic energy E w (0)). As another immediate consequence, if {w0 , w1 } and {u 0 , u 1 } both reside in B , then the energy E z (t) of their difference is uniformly bounded by 2CB . Proof The proof is standard and can be found in [13] (or [15]), and follows from the fact that  under the condition (6) on f , the perturbed energy Ew (t) remains bounded from below. 

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Corollary 2 (Auxiliary energy bounds) For any s0 ≤ s ≤ T0 on the trajectory T0 E z (s) ≤ E z (T0 ) +

˜ t) − D(z

s0

T0 ( f˜(z), z t )

(35)

s

and T0

T0 θ

( f˜(z(σ )), z t (σ ))dσ dθ

(36)

T0 T1 E z (θ )dθ + ( f˜(z(σ )), z t (σ ))dσ dθ

(37)

E z (θ )dθ ≤ (T0 − s)E z (s) + s

s

s

Likewise, for any s ≤ T0 ≤ T1 on the trajectory T0 (T0 − s)E z (T1 ) ≤ s

s

θ

Proof The first result readily follows from (34) after relabeling t in it by T0 . Then simply extend the (monotone) dissipation integral to the larger time-interval [s0 , T0 ] ⊇ [s, T0 ] forming an inequality. To prove (36) relabel “t” by “θ ” in (34) and integrate over θ ∈ [s, T0 ]. The same argument leads to (37) the third estimate, only now relabel s by θ , set t = T1 , and integrate for θ ∈ [s, T0 ].   To show that Ew (t) is a strict Lyapunov function (whence {H , S(t)} is a gradient system), the key needed result is the unique continuation property that allows to infer nullity of solutions that correspond to zero boundary velocity wt on geom × [0, T [ (for T sufficiently large). Such a unique continuation property follows from a slight modification of the argument given in [36], as shown in [47]. In fact, we have Proposition 4 (Strict Lyapunov Function) The energy functional Ew (t) is a strict Lyapunov function for the dynamical system {H , S(t)} corresponding to Eqs. 1 and 2. Thus {H , S(t)} is a gradient system. Proof First, must prove that Ew (t) is non-increasing, which readily follows from (33) since D(wt ) = χ(x)g(wt ), wt  ≥ 0 due to the monotonicity property of g. To complete the argument, it remains to check whether Ew (t) ≡ const for some generalized trajectory w implies that w is a stationary solution, i.e. wt = 0. Suppose Ew (t) is constant, then pick any approximating sequence of strong trajectories {w n , wtn } convergent

t

to {w, wt } in C([0, t], H ), then s  χg(wtn )wtn → 0 implying (by the monotonicity of g) that wtn → 0 pointwise a.e. in [s, t] × geom . From the strong convergence w n → w in C([s, t], H ) we have wtn → wt in H −1 s, t; H 1/2 () ; in addition, since wt ∈ L 2 (geom ) (follows from Theorem 1 and from (3)) conclude wt ≡ 0 a.e. on [s, t] × geom . Now we quote the unique continuation result in [47] (which, as was already mentioned above, is just a slight modification of the argument in [36]) to conclude that wt ≡ 0 in
, i.e. {w, wt } = {w, 0} is a stationary solution of (1).   3.2.2 Tangential Gradient The main technical tool for dealing with the trace dynamics will be the following estimate on the tangential component of the gradient trace (first established in [33, Lemma 7.2], and applied to a nonlinear setting in [14, p. 1859]). We present an adaptation of that result:

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Proposition 5 (Tangential estimates) Let {z, z t } be a strong solution to (19). Then there exists 0 < η < 1/2 so that for any α > 0, t > 0 one can find a constant Cα,η,t > 0 for which ⎫ ⎧ t+2α    t+α  2 ⎬ ⎨   2 ∂z ∇ z  ≤ Cα,η,t + z t2 d + [l.o.t. ]η (z) (38) tan ⎭ ⎩ ∂ν C α

{supp (h·ν)}∩

and t+α α

0

⎫ ⎧ t+2α    2 ⎬ ⎨   2 ∂z ∇ z  χ ≤ Cα,η,t χ(x) + z t2 d + [l.o.t. ]η (z) tan ⎭ ⎩ ∂ν  

(39)

0

 where ∇tan z = 2j=1 (∇z · θ j )θ j , and θ = {θ j } denotes a smooth orthonormal tangent frame on . The lower-order terms are given by [l.o.t. ]η (z) := z 21 +η,Q 2

t+2α

+ f˜(z) 2− 1 +η,Q 2

(40)

t+2α

and satisfy  [l.o.t. ]η (z) ≤ Cε0 ,B,η

t+2α

 z 2 + ε0

0

t+2α

∇z 2

(41)

0

Remark 8 Note that the tangential trace on the LHS of the inequality (38) is not a priori bounded in the finite energy space. In fact, even a formal application of the trace theorem would require “extra” 1/2 derivatives. This very special boundary behavior of the hyperbolic traces can be shown using the tools of microlocal analysis. Remark 9 (Weaker condition on the damping) By inspecting the details of the proof of Proposition 5 ([33, Lemma 7.2]) one concludes that the constant Cα,η,t can be made independent on t. This observation allows to yet further relax the assumptions on the damping in the proof of existence of a global compact attractor. In particular, to prove Part 2 of Theorem 2 it is sufficient to replace the condition (8) (and the aforementioned weaker version (9)), imposed on g with mere continuity of the damping along with the bounds m|s|2 ≤ g(s)s ≤ M|s|2 at infinity, i.e. for |s| > s0 > 0. Proof The estimates (38) and (39) follow as in [33, Lemma 7.2] with the following remarks: • Recall (e.g. see Proposition 2) that supp (h · ν) ⊂⊂ C so when repeating the localization argument in [36] we can construct a smooth cutoff whose restriction to the boundary is supported on C and equals 1 on supp (h · ν). 1 • To derive  the second estimate (39) we let χ˜ be any C extension of χ to
. Then the multiplier χ˜ can be naturally incorporated into the proof [33, Lemma 7.2] as a localizing cutoff in the space variable. It remains to verify the bound (41) on the lower order terms: t+2α 

z 21/2+η,Q t+2α

ε0 z + 2

≤ Cε0 0

123

t+2α 



2

0

∇z 2 + z t 2



(42)

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In order to estimate f˜(z) 2

− 21 +η,Q t+2α

, first pick φ ∈ H 1 (
). Use: the assumption (5),

two Hölder estimates (first choose conjugate exponents {3, 3/2}, then {4/3, 1/4}), and the Sobolev embedding H 1 (
) → L 6 (
) (in n = 3 dimensions) to derive:      1            ≤ C (1 + |w|2 + |u|2 )|z||φ|  ˜ = f (z)φ f (λz + u)zdλφ    


 0 1/3  2/3 2 2 3 3/2 3/2 ≤C (1 + |w| + |u| ) |z| |φ|

2/3 ≤ CB,
|z|3/2 |φ|3/2
    2/3 ≤ CB,





3/4

|z|(3/2)·(4/3)




≤ CB,
;2 z · φ 1,


|φ|(3/2)4

1/4

(where the constants CB,
and CB,
;2 provide a time-uniform bound on the finite energy w(t) 1,
+ u(t) 1,
, and depend on the diameter of the bounded set B from which we chose initial conditions {w0 , w1 }, {u 0 , u 1 }—see Corollary 1; the dependence of the constant on
stems from Sobolev embeddings). Therefore f˜(z) −1,
≤ const z L 2 (
) Proceed: f (w) − f (u) 2−1/2+η,Q t+2α ≤ Cε f˜(z) 2−1,Q t+2α + ε f˜(z) 20,Q t+2α  t+2α  ε0 t+2α ≤ Cε0 ,B z 2 + ∇z 2 ; 2 0 0

(43)

here the last step invoked the locally Lipschitz property (7) of the source and Poincaré’s inequality in order to bound f˜(z) by C ∇z . Now combine (42) with (43) to verify (41).   3.2.3 Pseudoconvex Function (x, t) Recall from Proposition 2 that the structure of the uncontrolled boundary geom permits to construct a scalar function d(x) with a positive definite hessian Hd ≥ ρ I . Pick a constant c  ρ 0 < c < min 1, (44) 2 Next let T be large enough to ensure cT > 2 max



y∈


|d(y)|

Let α be a (possibly small) parameter and define

(45)



T (x, t) = (x, t; α, T ) := d(x) − c t − α − 2

2 (46)

An important property of  implied by (45), is 2

∃ δ > 0 such that (x, α) = (x, α + T ) = d(x) − c

T < −δ 4

(47)

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In addition, one can always redefine d(x) (by merely adding a constant to it, since all the properties claimed in Proposition 2 will be retained) so that there exists subinterval [t0 , t1 ] with α < t0 < t1 < α + T (x, t) > 0 for t ∈ [t0 , t1 ]

(48)

Remark 10 In fact, all one needs is (x, t) > −δ for t ∈ [t0 , t1 ], however, since d(x) can be translated by an arbitrary value, it is more convenient to choose 0. Note that with the notation of Proposition 2 ∇ = ∇d = h henceforth the vector fields h(x) and ∇(x) will be used interchangeably. 3.3 Proving the Asymptotic Smoothness of the Flow The next lemma summarizes the main steps leading to the conclusion of Part 2 of Theorem 2. Lemma 1 (Compactness of the flow). Suppose the standing assumptions of Theorem 2 are satisfied. Let B denote a bounded set in the phase space H , and Y1 , Y2 ∈ B , with Y1 = {w0 , w1 } and Y2 = {u 0 , u 1 }. Recall the notation for the difference of the corresponding evolution trajectories: z = w − u. I. Fundamental identity I. Let the vector field h (its Jacobian denoted by Jh ) and the constant ρ > 0 be as defined in Proposition 2. If w and u are strong solutions then the following identity holds:   ρ (Jh − ρId)∇z · ∇z + (|∇z|2 + z t2 ) 2 t+α t+α Qα Qα    z t+α = Bα + f˜(z) h · ∇z + (div h − ρ) 2 t+α Qα  t+α z∇z · ∇div h − kα − 21  

(49)

t+α

Qα

where t+α

Bα

 := t+α

α t+α

kα

∂z 1 h · ∇z + ∂ν 2



=

t+α


z t h · ∇z



+ α

t+α

α

1 2

∂z 1 z(div h − ρ) + ∂ν 2








 z t2 − |∇z|2 (h · ν)

t+α

α

t+α z t z(div h − ρ) α

II. Observability inequality. There exists (large enough) t > 0 and (possibly small) α > 0 with T := t + 2α

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such that for any ε > 0 one can find a sufficiently large time s = mT + s0 (with 0 ≤ s0 < T ), m ∈ N for which s E z (s) ≤ ε + B (Y1 , Y2 ) ,ε,t,α

(50)

where s B (Y1 , Y2 ) := ,ε,t,α

m 

[(m− j)T,(m− j+1)T ]

B,ε,t,α

(Y1 , Y2 )

(51)

j=1

and mT +α+t [mT,(m+1)T ]

B,ε,t,α



(Y1 , Y2 ) := C1 sup z(mT + θ ) + C2



2

θ ∈[0,T ]




f˜(z)h · ∇z

mT +α ⎛ (m+1)T (m+1)T ⎞ (m+1)T mT  +t+α (m+1)T   θ   ⎠ ( f˜(z), z t ) dθ +C4 dθ −C5 ⎝ + + C3 mT +α (m+1)T



−C6

θ

mT

( f˜(z), z t ) + C7

mT

mT (m+1)T



mT +α

(52)

mT +α+t

( f˜(z), z t )

mT

where each coefficient Ci is dependent on B , ε, t, α in a continuous fashion (but independent of m). The observability inequality (50) is satisfied by all weak solutions. s III. Weak sequential compactness. Furthermore, the functional B (·, ·) in (50) satisfies ,ε,t,α s (Yn , Ym ) = 0 lim inf lim inf B ,ε,t,α m→∞

n→∞

(53)

for every sequence {Yk } in any bounded set B ⊂ H . The estimate (50) and the “weak sequential compactness” (53) of the functional  imply the asymptotic smoothness of the semigroup flow (see Theorem 4). Since, in addition, the system {H , S(t)} is of the gradient type (Proposition 4), and the Lyapunov function is bounded on bounded sets (Corollary 1), then by Theorem 3 the dynamics possesses a global compact attractor which coincides with the unstable manifold of stationary points, confirming part 2 of Theorem 2. Thus it remains to establish the validity of (50) and (53), which will be accomplished in Sect. 4. Remark 11 Inequality (50) will be derived from “first” Fundamental identity (49). Already t+α at this stage one notices that the boundary terms appearing in the expression Bα contain the gradient of z on the entire boundary. These terms cannot be fully absorbed by the energy or by the geometry, since a part of  has no geometric conditions imposed on it. It is here when tangential boundary estimates come into play; these inequalities allow to bound the tangential gradients of the solution by the dissipation and lower-order terms. However, the resulting inequality (50) may necessitate a longer observation interval in time. This last feature contributes to the “slicing” of time intervals appearing in the definition (51) of . Remark 12 The weak sequential compactness property (53) requires an estimate of the term f˜(z)h · ∇z = ( f (w) − f (u))h · ∇z. This product, via the divergence theorem leads to a boundary integral involving F(w) − F(u) where F is antiderivative of f , bounded by a polynomial of order 4 (when dim
= 3). This quantity is of the energy level, as follows

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from the critical Sobolev embedding H 1/2 () → L 4 (). Handling of this non-compact term will likewise rely on tangential boundary estimates. 3.4 The Fractal Dimension and the Regularity of the Attractor The next result lists milestones leading to a bound on the fractal dimension of the global attractor. Lemma 2 Suppose the assumptions of Theorem 2 are satisfied and let A be the global compact attractor (whose existence is verified by Part 2 of Theorem 2). Let B denote a bounded subset of H and Y1 , Y2 ∈ B , with Y1 = {w0 , w1 } and Y2 = {u 0 , u 1 }. As before set {z, z t } = {w, wt } − {u, u t }. I. Fundamental identity II. Pick any α ≥ 0, and let ρ, c, T , and (x, t), h be as defined above ((13), (45), (46)). Recall that Jh denotes the Jacobian of the vector field h (12). If {w, wt } and {u, u t } are strong solutions then for any τ > 0, the following identity holds for any ε0 > 0: ⎛ ⎞  ⎜ ⎟ T+α ⎜ eτ  (Jh − ρId)∇z · ∇z ⎟ ⎝ ⎠ + [Energy]α T+α

Qα

T+α

= (Bα )τ −τ

e

−τ 

(54)

 M21

f˜(z)M1

+

T+α

T+α

Qα

Qα

T+α

+ [Almost lower or der ] − (kα )τ where M1 = eτ  [h · ∇z − t z t ] T+α

[Energy]α



  eτ  |∇z|2 + z t2

:= (ρ/2 − c) T+α



 [Almost lower or der ] :=

z zt

Qα

$ % % µ $ρ d − ∇z · ∇ + f˜(z) − + c eτ  dt 2 2

T+α

Qα

with µ(x, t) := div (eτ  h) − ∂t (eτ  t ) and finally $

T+α

Bα

% τ

 :=

T+α

α

+

123

1 2



∂z M1 + ∂ν T+α

α

 T+α

α

(55)

% ∂z $ µ z − (ρ/2 + c)eτ  ∂ν 2

  eτ  z t2 − |∇z|2 (h · ν)

(56)

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T+α

(kα )τ :=

291



T+α  %T+α $µ eτ  z t (h · ∇z) + zt z − (ρ/2 + c)eτ  2

α α  T+α   τ 2 2 1 −2 e t z t + |∇z|


(57)

α

II. Intermediate observability inequality. Recall that the times t0 , t1 are such that (48) holds (for T as in (45), and small α > 0). Let B be any bounded subset of H with Y1 = {w0 , w1 } ∈ B , Y2 = {u 0 , u 1 } ∈ B and z = w − u. Then for T := T + 2α  T  T 2 ˜ t) + C E z (T ) + C T E z (θ )dθ ≤ C T ,α D(z T ,α sup z(θ ) 0 0 θ ∈[0,T ]  T  t1  T  θ ˜ +C1 dθ ( f (z), z t ) + C2 dθ ( f˜(z), z t ) (58) θ⎫ 0 0 ⎧t0 T T T ⎪ ⎪    ⎨ ⎬ ( f˜(z), z t ) − C4 ( f˜(z), z t ) −C3 + ⎪ ⎪ ⎩ ⎭ α

α+T

0

all the coefficients depend only on the parameters specified in respective subscripts, and on the diameter of B . III. Final observability inequality. Suppose, in addition, that B ⊆ A , i.e. the initial data comes from the attractor, then one can further refine (58): there exists 0 < σ < 1 such that E z (s + T ) ≤ σ E z (s) + C sup z(s + θ ) 2

(59)

θ ∈[0,T ]

for any sufficiently large T > 0 and any starting time s ∈ R (on the corresponding full evolution trajectory through the attractor). The estimate (59) essentially states that trajectories through the attractor converge exponentially to each other up to a compact perturbation. Once (59) has been verified, we can just appeal to the abstract “finite dimensional” criterion given in Theorem 5 to finish the proof of part 2 of Theorem 2. A similar argument was already used in the proof of [11, Theorem 2.2], and in [15, pp. 476–477]. In fact, it is possible to further refine the estimate and get an explicit bound on the fractal dimension [12, Sect. 2.2]. Moreover, inequality (59) also provides information on the regularity of the attractor: Corollary 3 Let Y0 = {w0 , w1 } ∈ A . Then the corresponding trajectory satisfies wtt + wt 1,
≤ CA where C depends only on the diameter of the attracting set in the state-space topology. Proof Apply (59) to z = w − u, where w(t) := u(t + h) for a small h > 0. Then dividing the relation by h 2 and using the uniform bound on the finite energy (dependent only on diam A , since u originates within the attractor) we get a uniform bound on the second time derivative and the gradient of velocity, namely on E u t (t), as h  0.   When combined with the original equation (1), Corollary 3 verifies Part 2 of Theorem 2. Let us once again emphasize the crucial technical difference between the estimate (50) of Lemma 1, used to deduce the existence of an attractor, and the inequality (59) of Lemma 2 which ultimately yields a bound on the fractal dimension and the regularity result. The

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“compensated compactness” condition (53) on the non-dissipative part  of (50) is much weaker than the genuine compactness of the seminorm H → R : {z, z t } → z L 2 appearing on the RHS of (59). While the former is weak in the sense that it permits iterated limits, the latter inequality ultimately requires decomposing the intrinsically non-compact contribution of the energy level terms of the form ( f˜(z), h · ∇z) and ( f˜(z), z t ) appearing in (54), into a compact semi-norm of z, plus some “absorbable” quantities; this obstacle stems precisely from the aforemetioned loss of compactness due to the critical exponent of the nonlinear map f (s). Consequently, verifying (59) is a more formidable task and it is at this point where we will resort to the Carleman estimates and will critically exploit the geometric structure of the attractor itself. Thus, in order to complete the proof of Theorem 2 it remains to verify the inequality (50), the condition (53), and the estimate (59) stated in the two lemmas above.

4 Existence of the Attractor (Proof of Lemma 1) 4.1 Fundamental Identity (49)—Part I of Lemma 1 Equation 49 is established by working with strong solutions, and applying, multipliers h · ∇z, and div (h)z to Eq. 19. Alternatively, (49) can be directly obtained from the much more general identity (54) by setting c = 0, t = 0 and τ = 0 in the latter. For brevity, rather than providing a (much simpler) direct derivation of (49) we limit ourselves to the general case of (54) which is fully derived in the appendix. 4.2 Energy Estimates: From (49) to the Observability Estimate (50)—Part II of Lemma 1 This section presents the steps leading from Part I (namely the identity (49)) of Lemma (1) to Part II of the lemma: the energy identity (50). Note that while the former holds only for strong solutions, the latter observability estimate is applicable to weak solutions as well. The presence of super-critical boundary traces (tangential gradients) arising from the limitations of the boundary damping is the main new component of the analysis.

4.2.1 The Geometry of
When the damping covers a subset of the boundary, the geometrical Assumption 3 makes it possible to construct a strictly convex function d(x) on
which was introduced back in Proposition 2. Essentially this condition ensures the absence of closed geodesics on
that do not effectively interact with the support C of the dissipation feedback [6]. From the strict convexity of d (see Proposition 2) it follows that the Jacobian of the field h = ∇d is strictly positive definite (see (13)), in particular with the aid of Poincaré’s inequality obtain ⎛ ⎞   t+α ⎜ ⎟ ρ ⎟ E z (t) ≤ ⎜ (J − ρId)∇z · ∇z + (|∇z|2 + z t2 ) Cρ h ⎝ ⎠ 2 α

123

t+α

Qα

t+α

Qα

J Dyn Diff Equat (2009) 21:269–314

293

Next, let T := t + 2α, apply the “pointwise” energy estimate (37) with s := α, T0 := α + t and T1 := T , and multiply both sides of the resulting relation by Cρ : 

t+α

t+α

tCρ E z (T ) ≤ Cρ

E z (θ )dθ + Cρ α

 dθ

α

T θ

( f˜(z), z t )

Combining the above two estimates conclude ⎛ ⎞  ⎜ ⎟ (Jh − ρId)∇z · ∇z ⎟ tCρ E z (T ) ≤ ⎜ ⎝ ⎠ t+α

Qα

t+α



ρ + 2

(|∇z| + 2

z t2 ) + Cρ

t+α Qα

T dθ

α

( f˜(z), z t )

(60)

θ

4.2.2 The Boundary Terms A crucial step in the argument is the estimate of the trace terms. As before, set T := t + 2α; then the following identity holds: t+α

Bα

T  ≤ Ct,α 0

T + ε0



  χ(x) z t2 + g(z ˜ t )2 + g(z ˜ t )z t

  E z (θ )dθ + ε0 E z (α) + E z (α + t)

0

+ Ct,α,ε0 ,B sup z(θ ) 2 θ ∈[0,T ]

(61)

This particular inequality is a simplified version Proposition 6, and can also be obtained from its statement directly by setting T = t and ignoring the parameter τ . Let us emphasize that this estimate entails the fundamental issue of bounding tangential gradients of the solution. Proposition 6 shows how to accomplish it by taking advantage of the trace estimates stated in Proposition 5.

4.2.3 “Almost Lower-Order” Terms Since the L 2 (
) norm of z resides below the corresponding finite energy level (H 1 (
)), the scalar products involving z can be estimated by an arbitrarily small multiple of the energy and “pure” lower order terms.

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z f˜(z) (div h − ρ)− 2

 z∇z · ∇div h ≤

1 2

t+α Qα

t+α Qα



  |∇z|2 + f˜(z)2 + Cε0 ,T

≤ ε0 t+α

sup

θ ∈[α,α+t]

z(θ ) 2

Qα

t+α ≤ ε0 CB

E z (θ )dθ + Cε0 ,T α

sup

θ ∈[α,α+t]

z(θ ) 2

(62)

In the last step in order to bound f˜(z) by the energy invoke the locally Lipschitz property of the source (7). 4.2.4 Energy at the End-Points of the Time Interval Finally, for the terminal products simply use the Schwartz and Poincaré inequalities: t+α

kα

≤ C (E z (α) + E z (α + t))

(63)

Note that C does not depend on t or α. 4.2.5 Combining the Estimates Substitute (60)–(63) into the fundamental identity (49); recall that T = t + 2α; after some relabeling of “inessential” coefficients (ε0 + ε0 CB by ε0 CB , etc.) obtain: T  tCρ E z (T ) ≤ Ct,α 0



  χ(x) z t2 + g(z ˜ t )2 + g(z ˜ t )z t

T + ε0 CB

  E z (θ )dθ + (C + ε0 ) E z (α) + E z (α + t)

0

 + Cε0 ,t,α,B sup z(θ ) + 2

θ ∈[0,T ]

t+α Qα

f˜(z)h · ∇z + Cρ

t+α

T dθ

α

( f˜(z), z t )

(64)

θ

 At this point some adjustment are needed to get rid of the energy integral ε0 CB

T

E z (θ )dθ

0

and the pointwise energy terms E z (α), E z (α + t) on the RHS. We also need to rewrite the damping terms via the energy identity (34). • In Corollary 2, inequality (36), set s = 0, T0 = T , multiply both sides by ε0 CB (i.e. the same parameter as the coefficient in the energy integral on the RHS in the above inequality)

T and add the result to (64). Then cancel ε0 CB 0 E z (θ )dθ on each side. Note that this step “pollutes” the RHS of the inequality by pointwise energy ε0 T CB E z (0). • Now invoke the inequality (35) with T0 = T , s0 = 0 and three different values for the third parameter: s = 0, s = α and s = t + α. As a result, (64) becomes (after some relabeling constant coefficients, e.g. C + ε0 by C since ε0 is bounded from above, etc.)

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tCρ E z (T ) ≤ Ct,α,B

T

0



295

  χ(x) z t2 + g(z ˜ t )2 + g(z ˜ t )z t



+(C + ε0 CB T )E z (T ) + Cε0 ,t,α,B supθ ∈[0,T ] z(θ ) 2 + t+α

θ

T

T +Cρ dθ ( f˜(z), z t ) + ε0 CB dθ 0 ( f˜(z), z t ) 0  αT θT



T + −C ( f˜(z), z t ) − ε0 CB T ( f˜(z), z t ) α

α+t

f˜(z)h · ∇z

t+α

Qα

(65)

0

Note that if we pick ε0 ≤ T −1 = (t + 2α)−1 then the coefficient (C + ε0 CB T ) of E z (T ) on the RHS is essentially independent of t (and T = 2 + 2α). Hence we may choose t large enough so that tCρ > C + ε0 T CB
C + CB Consequently the term E z (T ) on the right can be absorbed into the LHS after enough time passes since the beginning of the evolution. We then may drop E z (T ) on the RHS, and change the coefficient of E z (T ) on the left (possibly after a renormalization) to 1. • Finally, all of the dissipation and velocity traces in (65) appear with either a multiple of χ or have support in C (where χ is strictly positive). Hence, up to a multiple (dependent, besides the indicated parameters, on sup χ and inf C χ, which are fixed) we may invoke the assumption (8) on the damping, to conclude that for any η > 0, there exists positive Cη , WLOG Cη ≥ 1, such that T  Ct,α,B 0 energy ident.

≤



    2 2 χ(x) z t + χ g(z ˜ t ) + g(z ˜ t )z t ≤ Ct,α,B η + Cη

T

(34)

⎧ ⎨ Ct,α,B

⎩

T η + Cη (E z (0) − E z (T )) + Cη

˜ D(z t )

⎫ ⎬ ( f˜(z), z t ) ⎭

(66)

0

(note that the same result could have been derived by imposing the slightly weaker assumption (9) on the damping, as mentioned in Remark 3). Where the coefficient is rendered in bold just for ease of identification. The above discussion transforms (64) into (65), and then into: (Ct,α,B Cη + 1) E z (T ) ≤ Ct,α,B η + Ct,α,B Cη E z (0) + Ct,α,B sup z(θ ) 2 θ ∈[0,T ]  + f˜(z)h · ∇z t+α

⎫ ⎧ ⎛ T T⎞ t+α ⎪ T θ   T ⎪ ⎬ ⎨  T (67) ( f˜(z), z t ) + Cρ dθ +CB dθ −C ⎝ + ⎠ − CB ⎪ ⎪ ⎭ ⎩ Qα

α

θ

T

+Ct,α,B Cη

0

0

α

α+t

0

( f˜(z), z t )

0

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Where, we no longer need to track dependence on ε0 (can think of ε0 = T −1 for T = t + 2α and t large). Define σ :=

Ct,α,B <1 Ct,α,B + 1

Also recalling that we could have chosen (merely for convenience) Cη ≥ 1, whence Ct,α η/ (Ct,α Cη + 1) < η. Thus after dividing each side of the equation by (Ct,α Cη + 1) obtain [0,T ]

E z (T ) ≤ η + σ E z (0) + B,η,t,α (z)

(68)

The term  simply denotes all the integrals involving f˜(z) along with the square of the (lower-order) L ∞ (0, T ; L 2 (
)) norm of z. Furthermore, note that this inequality holds for all weak solutions since all the products are continuous with respect to the finite energy topology. Since all the norms are uniformly bounded with respect to the initial energy, this inequality is valid for any time interval (on the trajectory) of length T , in particular instead of 0 < α < t + α < T := t + 2α we could choose m ∈ N ∪ {0} and apply it to mT < mT + α < mT + t + α < (m + 1)T [(m−1)T,mT ]

which gives rise to the definition of B,η,t,α (z) in (52). Finally, in order to obtain (50) from (68) pick m ∈ N, then [(m−1)T,mT ]

E z (mT ) ≤ η + σ E z ((m − 1)T ) + B,η,t,α ...

≤

1−σ m 1−σ

z ((m

(z)

[(m−2)T,(m−1)T ]

[(m−1),mT ]

− 2)T ) + σ B,η,t,α (z) + B,η,t,α m [(m− j)T,(m− j+1)T ] m η + σ E z (0) + j=1 B,η,t,α (z)

≤ η + ση

+ σ2E

(z)

Setting s = mT + s0 where s0 < T we have E z (s) ≤ 2η + e(T

−1 ln σ )s

s e−1 E z (0) + B ,η,t,α

s where B simply denotes the sum of respective  over the T -intervals up to [(m−1)T, T ] ,η,t,α with m := s/T ; note that T is fixed. Since ln σ < 0, we can for any given ε > 0 choose, first, η sufficiently small (e.g. 4ε ), and then pick s large so that

2η + e(T

−1 ln σ )t

e−1 E z (0) < ε

Then in B,η,t,α term simply indicate the dependence on ε instead of η = η(ε). These steps confirm the statement (50) of Lemma 1. 4.3 The “Weak Sequential Compactness” of the Critical Terms (53)—Part III of Lemma 1 s It remains to prove that the functional B given by (51) and (52) satisfies the limit con,ε,t,α s [0,T ] dition (53). Since B,ε,t,α is given by a finite sum it suffices to analyze the case of B ,ε,t,α as given by (52) with m = 0. Let {w0k , w1k } ⊂ B be a sequence of initial data in a bounded subspace of H . Set z m,n := w m − w n . The lower order norms z m,n 2 appearing in B,ε,t,α contain a (uniform in time) Cauchy subsequence, as follows from the compactness of Sobolev’s embedding H 1 → L 2 and the fact that the trajectory w k ∈ L ∞ (0, T ; H 1 (
)) uniformly in k. Thus, some subsequence z m l ,nl 2 converges to 0 as l → ∞, whence lim inf m,n→∞ of these L 2 norms vanishes.

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The treatment of the terms involving ( f˜(z m,n ), z tm,n ) is identical to that in [15, Proposition a-11]. However, the integral  ( f˜(z m,n ), h · ∇z m,n ) t+α

Qα

requires a more careful analysis, than in the case of interior dissipation [15] due to the different boundary conditions. Step 1. Start with   m,n m,n ˜ = h · ∇(F(w m ) + F(w n )) f (z )h · ∇z t+α

t+α

Qα

Qα



−

h · [∇w n f (w m ) + ∇w m f (w n )] t+α

Qα



(h · ν)[F(w m ) + F(w n )]

= t+α

α

 (div h)[F(w m ) + F(w n )]

− t+α

Qα

 [(h · ∇w n ) f (w m ) + (h · ∇w m ) f (w n )]

− t+α

Qα

Note that for any weak solution {w, wt }, we have ∇ F(w) = f (w)∇w ∈ L 1 (
) whence by the virtue of the Sobolev embedding W 1,1 (
) → L 1 () (e.g. see [1, Theorem 5.22]) the above decomposition is justified. Since {w k , wtk } converges to some limit {w, wt } weakly∗ in L ∞ (0, T ; H 1 (
)× L 2 (
)), it is not hard to show (on a subsequence reindexed again by n) that f (w n ) converges weakly to f (w) in L ∞ (0, T ; L 2 (
)), and F(w n ) goes strongly to F(w) in L 1 (Q T ). See [15, pp. 498– 499] for more details. Consequently, passing to the limit and integrating by parts yields   lim inf lim inf f˜(z m,n )h · ∇z m,n = lim inf lim inf (h · ν)(F(w m ) n→∞

m→∞

n→∞

t+α

m→∞

t+α

α

Qα



+F(w n )) − 2

(h · ν)F(w) t+α

α

Thus to complete the proof of (53) it remains to show that (on a subsequence)   k ∗ ∞ 1 k w  w in L (0, T ; H (
)) ⇒ (h · ν)F(w ) → (h · ν)F(w) t+α

α

t+α

α

(69) where 0 < α < α + t < T .

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Remark 13 The above expression requires a passage to the limit in boundary terms that are critical with respect to the finite energy and, therefore, do not a priori form a pre-compact set. As mentioned before, tangential boundary estimates will be employed to handle this issue. Step 2. To verify (69) we need some additional estimates. Temporarily assume that we are dealing with strong solutions. Let ◦

γ :=supp (h · ν)⊂  where ◦ denotes the interior with respect to the topology of the (2D) manifold . Recall also that by construction γ ⊂⊂ C An application of (38) from Proposition 5, with z := v − 0 = v yields: t+α α

' '2 'v ' = 1,γ

t+α$ % ' '2 'v ' + ∇ v 2 tan γ γ α



T  ≤ Cα,T,B

C

0

∂v ∂ν

+ |vt |

T +Cε,α,T,B



2 2

T v 2


+ε

0

∇v 2
≤ . . . 0

applying the boundary and uniform bounds on the finite energy conditions continue as: T  . . . ≤ Cα,T,B



0

χ[g(vt )2 + vt2 ] + ε0 T CB

Since the damping is linearly bounded at infinity these terms can be estimated by a constant dependent on the initial energy. Conclude t+α v 21,γ ≤ CB,T

(70)

α

Step 3. Consequently if w and u are any two strong solutions we have 1



 (h · ν)[F(w) − F(u)] = t+α

α

(h · ν) t+α

0

α

t+α ≤C α

  f λw + (1 − λ)u dλ z d

γ



 1 + |w|3 + |u|3 |z| ≤ . . .

The integrand γ (1 + |w|3 + |u|3 )|z| via the embedding H 1 (
) → L 4 () (if dim
= 3) can be bounded uniformly in time by a constant dependent on the diameter of the set B

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299

where the initial data originated. Use this result to estimate the 2/3 power of the integral by a constant, thus continuing the above inequality as: ⎡ ⎤1 3 t+α  t+α 1     3 3 3 . . . ≤ CB ⎣ 1 + |w| + |u| |z|⎦ ≤ CB 1 + w L 6 (γ ) + u L 6 (γ ) z γ γ

α

α

⎞1/2 ⎛ t+α ⎛ t+α ⎞1/2  $  % ⎟ ⎜ ⎜ ⎟ 1 + w 2L 6 (γ ) + u 2L 6 (γ ) ⎠ ⎝ z 2/3 ≤ CB ⎝ ≤ ··· γ ⎠ α

α

Since  is a compact smooth two-dimensional manifold, we may take advantage of the embedding H 1 () → L p () for all 2 ≤ p < ∞, and invoke (70): ⎛ t+α ⎞1 2  1  t+α 1  t+α 2 2 2



  2 ⎜ ⎟ 2 2 3 . . . ≤ CB ⎝ z γ ≤ CB,T z 3 1 + w 1,γ + u 1,γ ⎠ α

α

(71)

α

In short: t+α α



(h · ν)[F(w) − F(u)] ≤ CB,T

⎛ t+α ⎞1 2  ⎜ 2/3 ⎟ ⎝ z γ ⎠

(72)

α

which is continuous with respect to the finite energy, hence holds for weak solutions as well. By compactness of the embedding H 1 (
) → L 4−ε () (in 3D), on a subsequence, z m,n converges strongly to 0 in L 2 (γ ) ⊂ L 2 (), thus the inequality (72) implies (69) and completes the proof of (53). This section completes the proof of Lemma 1 which, in turn, verifies, as was shown above in Sect. 3.3, Part 2 of Theorem 2.

5 Finite-Dimensionality of the Attractor (Proof of Lemma 2) The first step is to derive the fundamental identity (54) with Carleman weights. The calculations borrow some material from [15] and for the reader’s convenience the complete exposition can be found in the Appendix. However, special attention must now be paid to unbounded trace terms, which did not pose an issue for localized interior damping [15]. These quantities need to be tracked and properly estimated. 5.1 Energy Estimates: From (54) to the Intermediate Inequality (58)—Part II of Lemma 2 Until the end of this section we assume that all solutions are strong. The final result (58) extends to weak solutions by density and by the continuity of the relation with respect to the finite energy space.

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5.1.1 Energy on the LHS of (54) Recall that ρ/2 − c > 0 by construction (44). Since Jh is strictly positive definite (13), and eτ  ≥ 1 on [t0 , t1 ] (see (48)) we have (for a sufficiently small Cρ > 0) t1

T+α E z (t) + Cρ

Cρ t0

⎛ ⎜ eτ  E z (t) ≤ ⎜ ⎝

α



⎞ ⎟ eτ  (Jh − ρId)∇z · ∇z ⎟ ⎠

T+α

Qα



+(ρ/2 − c)

eτ  (z t2 + |∇z|2 )

(73)

T+α Qα

5.1.2 “Terminal” Energy Recalling (47) we get using Schwartz’ inequality   (kα,α+T )τ ≤ Ce−δτ E z (α) + E z (α + T )

(74)

5.1.3 The Observability Inequality for Boundary Dynamics Proposition 6 (Boundary estimates) Suppose the standing assumptions of Theorem 2 hold. For any positive parameters T , τ , α, ε0 we have $

T+α

Bα

% τ

T+2α

≤ Cτ,T ,α 0 T+2α

+ ε0



  χ(x) z t2 + g(z ˜ t )2 + g(z ˜ t )z t

  E z (t)dt + ε0 E z (α) + E z (α + T )

0

+ Cτ,T ,α,ε0 ,B

sup

z(θ ) 2

(75)

θ ∈[0,T +2α]

% $ T+α is given by (56). where Bα τ

Proof The strategy is to classify the traces according to the following categories: ˜ t ). When the damping is linearly bounded, this type of terms • Dissipation related, e.g.: D(z

2 2 may include χ(x)( g(z ˜ t ) + z t ), as well as the kinetic energy over the “controlled” 

2 segment C z t , or the kinetic energy accompanied by the cutoff:  χ z t2 . • Lower order terms: norms of z and z t in function spaces that embed compactly into resp. H 1 (
) and L 2 (
). In this case we will encounter norms of z in Sobolev spaces of order below H 1 (
) e.g. z 1−η,
, η > 0.

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301

T +2α • Small multiples of the total energy, e.g. ε0 0 E z (t)dt, ε0 E z (α), ε0 E z (T + α) etc.; these can be treated as almost lower-order provided the factor ε0 can be mollified at the expense of “genuine” lower-order norms. T+α

Now study each summand in (Bα )τ (56): I. Use the boundary condition (2), and the tangential estimates of Proposition 5 to derive T+α 

α

e

 τ

z t2

T+α

 2

− |∇z| (h · ν) =

e



α

τ



 z t2

−

 T+2α ≤ C T ,α,τ + z 1 +η,Q 2

2



C

0

T +2α

∂z ∂ν

2

 − |∇tan z| (h · ν) 2

 ˜ t )2 d z t2 + z 2 + χ g(z

+ f˜(z) 2− 1 +η,Q 2

(76)

T +2α

(Note that on C the term χ 2 can always be estimated by supC (χ)χ = Cχ) ∂z II. Recall that M1 = eτ  (h · ∇z − t z t ); estimate the summands in ∂ν M1 by using the boundary condition (20) and Proposition 5 T+α α



e

τ  ∂z

∂ν

T+α h · ∇z =



α

e

τ

T+α −



α



∂z ∂ν

(h · ν) − χ g(z ˜ t )(h · ∇tan z)

eτ  z h · ∇tan z  T+2α

≤ C T ,α,τ T+2α

+Ch



0

+ z 1 +η,Q

 z 2 + χ g(z ˜ t )2 d

  χ g(z ˜ t )2 + z t2 d

2

 



C

0

2



2

T +2α

+ f˜(z) 2− 1 +η,Q 2

T +2α

T+α

− α



eτ  z h · ∇tan z

(77)

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T+α 

α

e

τ  ∂z

∂ν

T+α t z t = −



α

e

τ

1 2

   

α

T+α − α



t z t z −



α

T+α

=−

T+α

eτ  t χ g(z ˜ t )z t

d  2 τ  d  τ  z e t − z 2 e t dt dt



eτ  t χ g(z ˜ t )z t



≤ Cτ,T ,α z(α) 2L 2 () + z(α + T ) 2L 2 () T+α +

T+α z(t) 2L 2 ()

+

α

˜ t) D(z

(78)

α

The next inequality was derived in [15, p. 487] and shows how to estimate the last integral on the RHS of (77): for any 0 < η < 1/2 we have T+α α



e

τ

z h · ∇tan z ≤ Cτ,α

T+α z 21−η,


(79)

α

which relies on the fact that  is a smooth compact manifold without boundary. III. Finally, recall that µ (55) is a smooth function whose supremum over
× [0, T ] ultimately depends only on τ and T (and α). Invoke the boundary conditions to get T+α α



% ∂z $ µ z − (ρ/2 + c)eτ  ≤ Cτ,T ∂ν 2

T+α α

, 

χ g(z ˜ t )2 + z 2

-

(80)

Remark 14 Note that in all of the preceding estimates we used the kinetic energy z t2 as a bound only in two cases: either when working over C , or when z t2 (on the boundary) is accompanied by cutoff χ. Under a full boundary damping such a strategy would not have been important; however, with geometrically restricted dissipation one has to control the ˜ t ) alone. Hence the damping is comparable only to χ z t2 , or to z t2  kinetic energy by D(z C (since inf χ > 0 on C ). Finally pick some (eventually small) constant ε0 > 0 and proceed as follows: (a) Apply inequalities (76)–(80) to (56). (b) Interpolate the lower order terms in the resulting inequality as follows: z(t) L 2 () ≤ z(t) 1/2+η,
≤ Cε0 z(t) + ε0 ∇z(t) Apply (41) to bound the term f˜(z) 2−1/2+η,Q from the inequality produced in T +2α step (a). Relabel the constants in the resulting estimates (in particular, without loss of generality, we can replace constant multiples of ε0 by just ε0 ) to recover the statement of Proposition 6.  

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5.1.4 The Damping The dissipation terms originating on the RHS of (75) can be handled by means of the uniform bounds (2) on the derivative of g: T+2α

T+2α



0

χ(z t2

+ g(z ˜ t ) + g(z ˜ t )z t ) ≤ C 2

˜ t) D(z

(81)

0

5.1.5 The Critical Product ( f˜(z)h · ∇z): the role of the parameter τ Criticality of the source makes it difficult to directly estimate the integral 

T+α




α

f˜(z)M1 =



T+α

α




f˜(z)(h · ∇z + t z t )

in the fundamental identity (54). In particular, the scalar products ( f˜(z), h · ∇z) and ( f˜(z), z t )

(82)

reside precisely at the level of the finite energy. Had the source been subcritical, we could have easily estimated these scalar products by sums of respective squares as ε z 21,
+ ε −1 (l.o.t. ), where l.o.t. denotes lower order norms, in particular those in spaces where the solution trajectories are compact, whereas the parameter ε could have been made arbitrarily small. However, such a decomposition is impossible since the embedding-dictated bound on f˜(z) is C
z 1,
, with no “regularity gap” to spare for Rellich-Kondrachov’s theorem. Also note that one cannot take advantage of the structure of the products in (82) (such as, for instance, replacing them by total derivatives), since f˜(z) = f (w) − f (u) and f is nonlinear. Thus quantities in (82) reflect the “essential” energy contribution of the source term. The interaction of the source with the kinetic component ( f˜(z), z t ) will have to be assessed using the information on the structure of the attractor itself (Sect. 5.2). Carleman estimates would not help here because terms of this sort intrinsically arise in the energy identity (34) and cannot be eliminated at the multiplier level. However, the influence of the source on the potential component ∇z must be handled by some other means, which is precisely where Carleman parameter τ comes into the picture:     −τ e−τ  M21 + e−τ  M21 + ε0 eτ  f˜(z)2 f˜(z)M1 ≤ −τ T+α

T+α

Qα

T+α

Qα

Qα

+ε0−1



T+α

Qα

e−τ  M21

T+α

Qα

Thus for τ > ε0−1 we have   e−τ  M21 + −τ T+α

Qα

T+α

Qα

f˜(z)M1 ≤ ε0



eτ  f˜(z)2

(83)

T+α

Qα

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Essentially, we have estimated the critical perturbation ( f˜(z), M1 ) in (54) by an arbitrarily small multiple of the energy (equivalent to f˜(z) 2 ). In order to relate f˜(z) and E z (t) we will also need the following inequality eτ /2 | f˜(z)|

1

≤

eτ /2

| f  (λw + (1 − λ)u)|dλ |w − u| ≤ C(1 + |w|2 + |u|2 )eτ /2 |z|

0

Observe that since the finite energy E w (t) and E u (t) is globally bounded (Corollary 1), we can replace the L 3/2 norm of w 4 and u 4 by CB . Consequently Hölder’s estimate and Sobolev’s imbedding H 1 (
) → L 6 (
) yield ' ' ' eτ  f˜(z)2 ≤ C '1 + w 4 + u 4 '0,3/2,
'eτ  z 2 0,3,
≤ CB eτ /2 z 21,
(84) ≤ CB eτ /2 ∇z 2 + CB,T,τ z 2 which can then be combined with (83) to produce  −τ

e

−τ 

 M21

T+α T+α τ /2 2 ∇z + CB,T,τ z 2 f˜(z)M1 ≤ ε0 CB e

+

T+α

α

T+α

Qα

Qα

(85)

α

5.1.6 Lower-Order Terms The estimates of [Almost lower order] terms in (54) are analogous to the argument in Sect. 4.2.3. Obtain    eτ  |∇z|2 + z t2 + f˜(z)2 + Cτ,T,ε0 sup z(θ ) 2 [Almost lower order] ≤ ε0 θ ∈[α,α+T ]

T+α

Qα

T+α ≤ ε0

(86) eτ  E z (t)dt + Cτ,T ,ε0

α

sup

θ ∈[α,α+T¯ ]

z(θ ) 2

where in the last step inequality (84) was used to obtain the weighted energy integral

T+α τ  e E z (t)dt on the RHS. α 5.1.7 Combining the Estimates Apply sequentially the estimates (73)–(75), (81), (85), and (86) to (54). As before, for convenience, define T := T + 2α then t1 Cρ t0

T+α T T τ ˜ E z (θ )dθ + (Cρ − ε0 ) e E z (t) ≤ Cτ,T ,α D(z t ) + ε0 CB E z (t)dt α

0

+ (ε0 + Ce

0

−δτ

  ) E z (α) + E z (α + T )

+ Cτ,T ,α,ε0 ,B sup z(θ ) 2 θ ∈[0,T ]

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Now (similarly to the procedure in Sect. 4.2.5): • To get a pointwise handle on the energy, in (37) set T1 = T ≥ T0 = t1 ≥ s = t0 : Cρ (t1 − t0 )E z (T ) ≤ Cρ

t1 t0

E z (θ )dθ + Cρ

t1

T dθ ( f˜(z), z t )

(88)

θ

t0

T • Analogously to the proof of existence of the attractor, the integral 0 E z (θ )dθ on the RHS of (87) needs to be eliminated. However, in this case, in order to deal with critical terms later in the proof, it will be useful to keep track of a bound on this integral, i.e. have it appear on the LHS of (87) instead. For that invoke (36) with s = 0 and T0 = T , and multiply the relation by 2ε0 CB : T

T E z (θ )dθ ≤ 2ε0 CB T E z (0) + 2ε0 CB

2ε0 CB 0

θ dθ

0

( f˜(z), z t )

(89)

0

• In order to bound the energy terms E z (0) (which originated in the first bullet above), E z (α), and E z (α + T ), invoke (35) with s0 = 0, T0 = T and respectively s = 0, s = α, and s = α + T . Summarizing: let ε0 ≤ Cρ , combine (87) with (88), add (89) to the result and cancel one

T instance of ε0 CB 0 E z (θ )dθ on each side. Then invoke (35) as described in the last bullet above. T Cρ (t1 − t0 )E z (T ) + ε0 CB

T E z (θ )dθ ≤ Cτ,T ,α,B

0

˜ t) D(z

0

  + 2(ε0 + Ce−δτ ) + 2ε0 CB T E z (T ) + Cτ,T ,α,ε0 ,B sup z(θ ) 2 θ ∈[0,T ]

t1 + Cρ

T dθ

t0

( f˜(z), z t ) + 2ε0 CB

θ

− (ε0 + Ce−δτ )

⎧ T ⎨ ⎩

α

T

θ

0

0

dθ ( f˜(z), z t )

⎫ T ⎬ T + ( f˜(z), z t ) − 2ε0 T ( f˜(z), z t ) ⎭ α+T

(90)

0

Now choose, ε0 small (! CB T ), and τ large so that 2(ε0 CB T + Ce−δτ ) + 2ε0 T ! Cρ (t1 − t0 ) Then the resulting small multiple of the energy E z (T ) can be absorbed into the LHS, leading to the intermediate observability inequality (58). Note that since the strong solutions converge strongly to weak trajectories in C([0, T ]; H ), this inequality holds for weak solutions as well. At this stage one can fix ε0 and no longer keep track of its influence on the parameters. This step completes Part II of Lemma 2.

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5.2 Final Estimates and the Critical Product ( f˜(z), z t )—Part III of Lemma 2 Rewrite the dissipation terms in the intermediate observability estimate (58) using the energy identity (34). Obtain: T (C T ,α + 1)E z (T ) + C T

E z (θ )dθ ≤ C T ,α E z (0) + C T ,α supθ ∈[0,T ] z(θ ) 2 0

+C T [Critical terms]

where [Critical terms] incorporates all integrals involving ( f˜(z), z t ) from the RHS of (58)

T and C T ,α 0 ( f˜(z t ), z t ) which just appeared after rewriting the damping terms via the energy identity (34). After dividing both sides by (C T ,α + 1) arrive at a version of the observability estimate (59), but perturbed by critical source dependent terms: T E z (T ) + C T ,α

E z (θ )dθ ≤ σ E z (0) + C T ,α sup z(θ ) 2 + C T ,α [Critical terms] θ ∈[0,T ]

0

(91) where σ =

C T ,α,ε0 C T ,α,ε0 + 1

<1

Thus it remains to analyze the critical interaction of the source with the velocity: ( f˜(z), z t )

(92)

for trajectories through the attractor (since that’s all we need for Part III of Lemma 2). Recall that another critical product: ( f˜(z), h · ∇z), was originally handled via the Carleman estimates with a large parameter τ , however (92) is intrinsic to the basic dissipative law (34) and cannot be eliminated with multipliers alone. To deal with (92) we can employ the method originally introduced in [29] to study von Karman equation with internal damping, and then later used for boundary-damped wave (see [10]) and von Karman (see [11]) equations. See also [12] for an abstract realization of this idea. Full details of this argument are identical to those presented in [15]. Here we will only summarize the main points of the proof. The approach is based on the geometry and the compactness property of the attracting set. The argument first establishes that every trajectory through the attractor is a strong trajectory. This fact can be deduced by setting z := w(t + h) − w(t) and tracing the full trajectory {w(t), wt (t)} backward in time (t → −∞) to a neighborhood of a stationary point (as follows from the structure of the attractor). There the velocity component is small (hence ( f˜(z), z t ) is small as well) producing from (91), after some work: T E z (s + T ) + C T

E z (θ )dθ ≤ σ E z (s) + C T sup z(s + θ ) 2 , s + T < Cw s

θ ∈[0,T ]

where Cw , could be negative (we are working on a full trajectory), and depends on the specific solution w through the attractor; however, all the constant parameters depend only on the relative length T of the interval [s, s + T ], not on the absolute time s, and on the diameter of the attractor in the finite energy space. Normalizing by h 2 one can obtain bounds on the time

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307

derivative of the energy ∇wt + wtt , concluding eventually via forward well-posedness that the attractor is a subset of H 2 × H 1 (at this stage we do not claim yet that this subset is bounded in the higher topology). Thus, the attractor A due to the compactness can be approximated by finitely many points in the energy space H , however each of those “nodes” lives in H 2 × H 1 . This gain in regularity helps to realize that ( f˜(z), z t ) can, roughly speaking, be treated as

T

( f˜(ˆz (t)), z t ) + ε

0

T z t 2 0

where zˆ (t) is one of the finitely many (!) H 2 nodes, hence bounded in H 2 independently of t and rendering the product subcritical. And the error ε depends on the maximal H -distance between the nodes of the set approximating the attractor, and can a priori be made arbitrarily small due to the compactness of the latter. Thus ( f˜(z), z t ) can be expressed as a subcritical product plus an arbitrarily small multiple of the quadratic energy. The latter

T will be absorbed into the integral C T 0 E z (θ )dθ on the LHS of (91), producing the desired observability estimate (II). A detailed exposition of analogous argument can be found in [15, Sect. 6.8]. This argument completes the last part of Lemma 2, which in turn, as was shown above, verifies the parts 2 and 2 of Theorem 2.  

A Appendix: The Fundamental Identity (54) In this section we will verify the extended fundamental identity of Lemma 2. The result follows if we study the action of the strong formulation (19) of the original problem on specially constructed multiplier maps. The argument below is a modification of the one in [15], which in turn was inspired by Lasiecka and Triggiani [34]. The reader may consult [15] for yet another detailed and self-contained exposition. To keep notation compact we will be working over the time-interval [α = 0, T = T ], with respect to the notation introduced in Lemma 2. This substitution does not affect the argument and the change of terminal points is straightforward. Pick any τ > 0, and let ρ, c, T , and (x, t), be as defined respectively in (13), (45), (46). Recall that h = ∇, and Jh denotes the Jacobian of this vector field.   A.1 “Flux” Multiplier M1 = eτ  h · ∇z − t z t Define: 

M1 = M1 (, τ ; z, z t ) := eτ  h · ∇z − t z t



(A.93)

Use M1 as the test function in the strong formulation of (19), and integrate the relation in time. Expand the result term by term:

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Product with z tt   z tt eτ  [h · ∇z − t z t ] = QT

1 2

eτ  z tt h · ∇z − QT

 =

eτ  z t h · ∇z









e τ  t QT

−

eτ  t z t2

zt QT

0



1 − 2

T



T + 0

1 2



d  2 z dt t

 d  τ e h · ∇z dt

QT

z t2

d  τ  e t dt

(A.94)

Let  k1;0,T (z, z t ) :=

In addition  QT

e




 d  τ e h · ∇z = zt dt

τ

 zt

1 h · ∇z − t z t 2

T (A.95) 0



z t h · (τ eτ  t ∇z + eτ  ∇z t )   1 eτ  z t t h · ∇z + eτ  z t2 (h · ν) =τ 2 T QT  1 − z 2 div (eτ  h) 2 QT t QT

(A.96)

Substitute the last relation into (A.94):      1 2 d τ τ z tt M1 = z eτ  z t t h · ∇z (e t ) + div (e h) − τ 2 Q T t dt QT QT  1 eτ  z t2 (h · ν) + k1;0,T (A.97) − 2 T Product with −z    ∂z (−z)eτ  [h · ∇z − t z t ] = − M1 + τ (∇z · h)M1 QT T ∂ν QT   + eτ  ∇z · ∇(h · ∇z) − t eτ  ∇z · ∇z t (A.98) QT

QT

Now recall that Jh denotes the Jacobian of h, and invoke the identity   1 ∇v · ∇(h · ∇v) = Jh ∇v · ∇v + h · ∇|∇v|2 2 to obtain



123

 1 eτ  h · ∇|∇z|2 2 QT QT   1 = eτ  (Jh ∇z) · ∇z + eτ  |∇z|2 (h · ν) 2 QT T  1 τ 2 − div (e h)|∇z| (A.100) 2 QT

eτ  ∇z · ∇(h · ∇z) = QT



(A.99)

eτ  (Jh ∇z) · ∇z +

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Next, 

t eτ  ∇z · ∇z t = QT

1 2



d 1 |∇z|2 = dt 2  d  |∇z|2 t e τ  dt

t e τ  QT

1 − 2



QT




t eτ  |∇z|2

T 0

(A.101)

Introduce another shorthand for the terminal products  k2;0,T := 21




t eτ  |∇z|2

T (A.102) 0

Equations (A.100)–(A.102) permit to rewrite (A.98) as 

 (−z)M1 = − QT



T

+ QT



1 ∂z M1 + ∂ν 2



eτ  |∇z|2 (h · ν)     1 d  t e τ  eτ  (Jh ∇z) · ∇z − |∇z|2 div (eτ  h) − 2 QT dt

+τ

T

(h · ∇z)M1 − k2;0,T

(A.103)

QT

Result of the Multiplication by M1 Put together (A.97) and (A.103) to establish that when the strong formulation of (19) acts on the test function M1 = eτ  [∇ · ∇z − t z t ] = eτ  [h · ∇z − t z t ]

it produces the identity:    ∂z 1 M1 + eτ  z t2 − |∇z|2 (h · ν) ∂ν 2 T T    2  d τ  2  1 1 z t + |∇z|2 z − |∇z|2 div (eτ  ∇) = (e t ) + 2 QT dt 2 QT t  eτ  (Jh ∇z) · ∇z + QT   +τ (h · ∇z)(M1 − eτ  z t t ) − f˜(z)M1



QT

QT

+k1;0,T − k2;0,T

(A.104)

A.2 Equipartition of Energy: Multiplier M2 = z m(x, t) Let m(x, t) be some C 1 function defined on Q T . Later we will specify several useful candidates for m(x, t). Introduce the second multiplier M2 (m) := zm

(A.105)

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Multiply Eq. 19 by M2 (m) and integrate over the space–time cylinder Q T . Some of the terms in the resulting inequality can be expanded as follows:  T   z tt zm = z t zm − (z t2 m + z t zm t ) QT
Q 0   T  ∂z zm − (|∇z|2 m + z∇z · ∇m) (z) zm = QT T ∂ν QT Hence the second multiplier yields (after some rearrangement of the terms)      ∂z zm + [z t2 − |∇z|2 ]m = − z(∇z · ∇m − z t m t ) ∂ν QT T QT T   z t zm − f˜(z)zm +


QT

(A.106)

0

A.3 Isolating Energy-Level Terms Define µ(x, t) := div (eτ  h) −

d dt



e τ  t



(A.107)

Substitute m = µ(x, t) into the identity (A.106), then multiply the expression by 21 and rearrange:     2   2  d  τ  1 1 ∂z 1 e t − z t − |∇z|2 div (eτ  h) = z t − |∇z|2 zµ 2 QT 2 QT dt 2 T ∂ν  1 z (∇z · ∇µ − z t µt ) + 2 QT  T  1 1 − z t zµ (A.108) f˜(z)zµ + 2 QT 2
0 Let k3;0,T := k1;0,T − k2;0,T +

1 2




T z t zµ

(A.109)

0

Recalling the definition (A.93) of M1 we have 

eτ  (τ 2t −2c)

. /0 1  2 d zt (h · ∇z)(M1 − eτ  z t t ) (eτ  t ) +τ dt QT QT   τ 2 2 e z t t + τ e−τ  (M1 + eτ  z t t )(M1 − eτ  z t t ) =τ QT QT  −2c eτ  z t2 QT   =τ e−τ  M21 − 2c eτ  z t2 (A.110) QT

QT

Substitute (A.108) into (A.104), then apply (A.109) and (A.110) to the result. In addition

add and subtract the term Q T ρeτ  |∇z 2 | on the RHS of the resulting equation; as far as the

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fundamental identity is concerned the value of ρ is irrelevant; however, eventually we will use the one stated in relation (13). We finally arrive at 

    ∂z ∂z 1 1 M1 + eτ  z t2 − |∇z|2 (h · ν) zµ + 2 T ∂ν 2 T T ∂ν  = eτ  (Jh − ρId)∇z · ∇z QT   ρeτ  |∇z|2 − 2c eτ  z t2 + QT QT   1 + z(∇z · ∇µ − z t µt ) + τ e−τ  M21 2 QT QT  % $ zµ + M1 − f˜(z) 2 QT +k3;0,T

(A.111)

A.4 Reconstructing the Quadratic Energy from the Potential Component The next step reconstructs “almost full” quadratic energy ∇z 2 + z t 2 (short of the lower order term z 2L 2 () ) from just the potential component ∇z 2 via the equipartition relation.

Begin with (A.111) and split the newly added term Q T ρeτ  |∇z 2 | into two summands:  ρ

e

τ

 |∇z| − 2c 2

QT

QT

eτ  z t2

 = (ρ/2 − c) QT

eτ  (|∇z|2 + z t2 )



+(ρ/2 + c) QT

eτ  (|∇z|2 − z t2 )

(A.112)

Invoke the equipartition identity (A.106) with m = −eτ  :  QT

(|∇z|2 − z t2 )eτ  =



 T



+



  d z ∇z · ∇eτ  + z t (eτ  ) dt QT T  z t zeτ  (A.113) f˜(z)zeτ  −

∂z ∂ν

zeτ  −






QT

0

Substitute (A.113) in place of the corresponding integral on the RHS of (A.112). Then use

the outcome to replace the difference ρ Q T eτ  |∇z|2 − 2c Q T eτ  z t2 in (A.111). Before stating the result group together some of the terms:    ∂z ∂z 1 z µ − (ρ/2 + c)eτ  M1 + 2 T ∂ν T ∂ν    1 + eτ  z t2 − |∇z|2 (h · ν) 2 T T  := k3;0,T − (ρ/2 + c) z t zeτ  

(B0T )τ :=

k0,T




(A.114) (A.115)

0

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We have:  (B0T )τ =

QT

eτ  (Jh − ρId)∇z · ∇z + (ρ/2 − c)



+

z∇z · ∇ 

QT

−

z zt Q

T +τ  +

QT

QT

+k0,T

$µ

τ

%

 QT

eτ  (|∇z|2 + z t2 )

− (ρ/2 + c)e 2 % d $µ − (ρ/2 + c)eτ  dt 2

e−τ  M21 %  $µ f˜(z) M1 + z − (ρ/2 + c)eτ  2 (A.116)

A mere rearrangement of the summands in (A.116) and replacing the interval [0, T ] with [α, α + T ] yields (54). Acknowledgements The authors would like to express their gratitude to the referee whose constructive suggestions helped significantly improve the clarity and quality of this paper. Research of I. Lasiecka has been supported by NSF Grant DMS 0606682.

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