DISCRETE AND CONTINUOUS Website: http://aimSciences.orgDYNAMICAL SYSTEMSVolume 20, Number 3, March 2008 pp. 459–509

LONG-TERM DYNAMICS OF SEMILINEAR WAVE EQUATION

WITH NONLINEAR LOCALIZED INTERIOR DAMPING AND A

SOURCE TERM OF CRITICAL EXPONENT

Igor Chueshov

Department of Mathematics and Mechanics, Kharkov National UniversityKharkov, 61077, Ukraine

Irena Lasiecka

Department of Mathematics, University of VirginiaCharlottesville, VA 22904, USA

Daniel Toundykov

Department of Mathematics, University of Nebraska - LincolnLincoln, NE 68588, USA

Abstract. This article addresses long-term behavior of solutions to a semilin-ear damped wave equation with a critical source term. A distinctive feature ofthe model is the geometrically constrained dissipation: it only affects a smallsubset of the domain adjacent to a connected portion of the boundary. Themain result of the paper provides an affirmative answer to the open questionwhether global attractors for a wave equation with critical source and geomet-rically constrained damping are smooth and finite-dimensional. A positiveanswer to the same question in the case of subcritical sources was given in [9].However, critical exponent of the source term combined with weak geometri-cally restricted dissipation constitutes the major new difficulty of the problem.To overcome this issue we develop a new version of Carleman’s estimates andapply them in the context of recent results [12] on fractal dimension of globalattractors.

1. Introduction.

1.1. The problem. Let Ω ⊂ Rn=3 be a smooth bounded connected domain withconnected boundary Γ. By “smooth domain” we will henceforth understand anonempty open set that locally lies on one side of its boundary, and the boundaryforms a compact orientable C2 manifold. Let QT := Ω×]0, T [ and ΣT := Γ×]0, T [.This paper addresses long-term behavior of solutions to the following semi-linearwave equation:

wtt − ∆w + χ(x)g(wt) = f(w) in QT

w(0) = w0, wt(0) = w1 in Ω.(1)

2000 Mathematics Subject Classification. Primary: 35B41; Secondary: 35B33, 35L70.Key words and phrases. wave equation, attractor, fractal dimension, localized damping, non-

linear damping, source, critical exponents.Research of the second author is partially supported by NSF Grant DMS-0606682.

459

460 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

Nonlinear maps g and f represent Nemytski operators associated with scalar con-tinuous real-valued functions g(s) and f(s) respectively. Feedback map f modelsthe source term, and F will denote the respective anti-derivative

F ′(s) = f(s), F (0) = 0. (2)

Map g is monotone increasing and represents interior dissipation. The nonnegativecutoff function

χ ∈ L∞(Ω; R+ := [0, ∞[

)(3)

potentially restricts the action of g; we are primarily interested in the case whenthe support of χ covers only a subset of a collar of Ω. More precisely: χ may haveany configuration as long as there exists a nonempty segment of the boundary ΓC

(for “Controlled”), and some fixed θ > 0 such that

Ωχ :=x ∈ Ω : dist (x, ΓC) ≤ θ

⊆ suppχ (4)

and χ has a positive a.e. uniform bound from below on Ωχ:

ess infx∈Ωχ

(χ(x)) > 0.

Remark 1. There is no restriction on the “thickness” θ > 0 of the dissipative layerin (4).

The remaining portion of the boundary (possibly overlapping with ΓC) is denotedΓU (see Figure 1 below) signifying the “Unobserved” segment around which thedissipation is (mostly) absent:

Γ = ΓC ∪ ΓU (5)

Segment ΓC is assumed to be open connected and have positive measure. If ΓC 6= Γthen the same assertions apply to ΓU , otherwise take ΓU = ∅. When nonempty,ΓU will overlap with ΓC since these sets form an open cover of the connected set Γ.In addition ΓU will have to satisfy some additional geometrical assumptions thatdepend on the boundary conditions and will be stated later.

The main goal of the paper is to establish finite-dimensionality of global attrac-tors associated with finite energy solutions to equation (1) with the source f ofcritical exponent. It is known that criticality of the source in hyperbolic problemsconstitutes the major difficulty when investigating properties of attractors becauseof the loss of compactness. The classical tools developed for the study of attrac-tors are no longer applicable in this setting. In fact, existence of finite-dimensionalattractors for a “critical” wave equation with nonlinear damping supported on theentire domain has been shown only recently [11].

Problems where the damping is geometrically constrained (be it either boundaryor localized near the boundary) are much more subtle, as the issue of geometry andwave propagation comes into play. More so, in such a setup the critical behavior ofthe source must be handled at the level of both kinetic and potential components ofthe energy (respectively the time and space derivatives of w), as opposed to, for in-stance, Petrowski-type systems (e.g. plate-like, Schrodinger) where the interactionof the source with the gradient of the displacement (acoustic pressure) is inher-ently of a lower-order. For these reasons the up-to-date results on dimensionalityof attractors for the wave equation with geometrically restricted dissipation focusexclusively on subcritical sources [9].

The main contribution of this paper is showing that even in the presence of acritical source a damping localized on a small patch near the boundary can provideboth: finite-dimensional behavior and smoothness of asymptotic regimes. These

LONG-TERM DYNAMICS OF A WAVE EQUATION 461

properties are not necessarily expected in hyperbolic-like problems where the insta-bility is of inherently infinite-dimensional nature.

To our best knowledge this is the very first result pertaining to finite-dimensionalityand smoothness of attractors for a wave equation with critical exponent and dissi-pation of restricted support.

Furthermore, our method accommodates mixed Dirichlet/Neumann boundaryconditions. The Neumann setting is a significant obstacle in its own right becausesuch a system no longer satisfies the Lopatinskii condition. To our knowledge thereare no results predating [35, 46] which address asymptotic energy behavior of lo-cally damped hyperbolic systems with Neumann boundary data. Thus, additionalnovelties of the present exposition include: localized nonlinear damping, Criticalsource term, and Neumann-type mixed boundary conditions.

The technical aspects of our analysis are based on (i) Carleman estimates devel-oped for nonlinear problem with non-Lopatinskii-type boundary; and (ii) suitableobservability inverse-type estimate, which can be applied to dynamics that evolvesbackwards in time (i.e. extended to the negative time-scale).

1.2. Basic notation. The bilinear form (·, ·) will denote the inner product inL2(Ω). To avoid confusing inner-products and ordered pairs we will slightly abuseset-theoretic notation using a1, a2, . . . , an to indicate an ordered n-tuple“(a1, a2, . . . , an).” Pairing [[·, ·]] will denote a commutator.

Besides the standard notation for Sobolev norms we shall use some additionalconventions:

• ‖w‖s,p,X to indicate the norm in the Sobolev space W s,p(X).• ‖w‖s,X as a shorthand for the norm in Hs(X)• ‖w‖ := ‖w‖L2(Ω)

To study asymptotic behavior of the solution w(t), wt(t) to system (1) we willneed to examine differences of evolution trajectories, hence the following notationwill appear frequently:

z := w − u

g(zt) = gu(zt) := g(zt + ut) − g(ut) = g(wt) − g(ut) (6)

f(z) = fu(z) := f(z + u) − f(u) = f(w) − f(u). (7)

From (1) it follows that z should satisfy

ztt − ∆z + χg(zt) = f(z) in QT

z(0) = w0 − u0, zt(0) = w1 − u1 in Ω.(8)

Reserve a few more symbols for the dissipative terms:

D(wt) := (χg(wt), wt), D(zt) := (χg(zt), zt).

Let us also preview some of the functions that will appear later in the paper:

• Smooth cutoff maps ξ, ζ : Ω → [0, 1] will be used to partition domain Ω intodissipative and non-dissipative (where χ(x) ≡ 0) regions. Operators Mξ andMζ will respectively denote pointwise (a.e.) multiplication by ξ and ζ.

• Certain vector field h will be explicitly constructed from the information onthe geometry of the unobserved boundary ΓU ; and will help estimate thetraces on the portion of the boundary where the damping is absent.

462 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

A precise definition of ξ, ζ and h will be given in Sections 1.4.1, 1.4.2. Throughoutthe article we will repeatedly use symbol C for asymptotic notation Θ(1), i.e. C willindicate a “generic” constant whose precise algebraic transformations throughoutthe estimates are not essential, hence will not be written out explicitly. Sometimes,for clarity, we will indicate the dependence of the undetermined constants on otherquantities by using suggestive subscripts.

1.3. Assumptions on the nonlinear terms. Existence and structure of globalattractors critically depend on the behavior of the source and the dissipation. Wesubject the maps f and g to the following conditions:

Assumption 1 (Source). The source term is modeled by f ∈ C2(R) satisfying

1. In 3-dimensional case let map f be of (at most) the critical (cubic) order.This assertion follows from the following assumption:

|f ′′(s)| ≤ Cf (1 + |s|) (9)

2. The following supremum is finite:

lim sup|s|→∞

s−1f(s) < λ0 (10)

where λ0 > 0 is the minimal eigenvalue of the elliptic spectral problem

−∆u = λu in Ωu satisfies the boundary conditions (B.C.-1, 2, or 3) (Section 1.4)

Assumption 2 (Damping). Let g ∈ C(R) be monotone increasing with g(0) = 0.Assume there exists sg > 0 and constants m, M > 0 such that

m|s| ≤ |g(s)| ≤ M |s| for |s| ≥ sg. (11)

Remark 2. When dimension of Ω equals 2 the space H1(Ω) embeds into Lp(Ω) forany p ≥ 1, hence assumption (9) may be replaced by any polynomial growth. Thetwo-dimensional case is much easier to analyze and will not be dealt with separately.Higher-dimensional domains Ω ⊂ Rn>3 can be treated similarly by imposing theappropriate (critical) growth condition on the map f . We opt for the 3D framework,which is a benchmark problem representative of all the difficulties related to thestudy of critical exponents.

Proving finite-dimensionality of the attractor under localized dissipation will re-quire a stronger condition on the damping:

Assumption 3 (Only for finite-dimensionality and smoothness). Let g ∈ C1(R)with g(0) = 0. We assume there exist positive constants m and M such that

m < g′(s) < M s ∈ R. (12)

Due to (9) and basic Sobolev embedding results, function f corresponds to alocally Lipschitz Nemytski operator H1(Ω) → L2(Ω). In particular

‖f(w) − f(u)‖ ≤ C(r)‖w − u‖1,Ω ∀ ‖w‖1,Ω, ‖u‖1,Ω ≤ r < ∞ (13)

The order of the source can go up to the critical level (O(x3) for Ω ⊂ R3) allowedby the well-posedness theory. Critical threshold is determined from Sobolev embed-dings: it corresponds to the largest admissible exponent (in the polynomial boundon f) for which f : H1(Ω) → L2(Ω) is bounded.

LONG-TERM DYNAMICS OF A WAVE EQUATION 463

Sources of higher order raise the issue of well-posedness of the solutions in thespace of finite energy (w ∈ H1(Ω), wt ∈ L2(Ω)), unless the damping is suitablystructured to counteract the influence of the source. Thus, restriction of nonlinearityup to the critical exponent is a natural one when studying long-term behavior.Furthermore, existence of attractors is strictly linked with asymptotic compactnessof trajectories, while at the critical level the compactness of Sobolev embeddingsis lost, thus critical growth poses major difficulties that call for techniques entirelydifferent from those employed in subcritical cases.

Due to restricted support of the dissipation, linear bounds at infinity in (11) arenecessary to establish existence of the attractor. Counterexamples are known in 1-dimensional cases: e.g. [47] for the boundary damping (which poses same technicalissues as the localized interior dissipation). The authors of [47] prove that withouta linear bound at infinity, the uniform decay of the energy necessarily depends onhigher-order norms of the solutions. In particular, such decay requires initial stateto be more regular than the finite energy level. Detailed exposition on how sub-and super-linear damping at infinity correlates the decay with higher-order normscan be found in [35].

As far as finite-dimensionality is concerned, the question whether one can relaxcondition (12) near the origin still remains open in the context of localized dissi-pation; in fact, even under full interior damping it is unknown whether the lowerbound g′(0) > 0 can be dispensed with.

1.4. Boundary conditions and geometry of the domain. Our approach canaccommodate different types of boundary dynamics. Depending on the nature ofthe boundary conditions (b.c.) it may be necessary to impose certain assumptionson the shape of Ω, which later resurfaces in the construction of multipliers for theenergy estimates.

Assumption 4 (Basic geometrical condition). If there exists an unobserved seg-ment of the boundary, i.e. ΓU 6= ∅, then we assert existence of a point x0 ∈ Rn

such that

(x − x0) · ν(x) ≤ 0 on ΓU (14)

with ν being the outward normal vector field on Γ.

To present mixed boundary conditions we will we need another partition of Γ:

Γ = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅

both segments Γi being relatively open, connected, having positive measure, withcontinuous (1-d) boundary. Henceforth let Σ0

T := Γ0×]0, T [, Σ1T := Γ1×]0, T [,

and define the interface set

J := Γ0 ∩ Γ1. (15)

We will consider the following cases:

(B.C.-1) (Dirichlet) w∣∣ΣT

≡ 0

(B.C.-2) (Dirichlet/Neuman) For a function k1 ∈ C1(Γ1; R+

),

w∣∣Σ0

T

≡ 0,

(∂w

∂ν+ k1w

) ∣∣∣∣Σ1

T

≡ 0.

Here we require J (15) to fall within the dissipative region, away from ΓU :J ⊂ int(ΓC \ΓU ) (where “int” stands for “interior” in Γ). Note that under

464 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

this assertion we either have

Γ0 ⊂ ΓC \ ΓU (16)

or

Γ1 ⊂ ΓC \ ΓU . (17)

(B.C.-3) (Neumann/Robin) Suppose at least one of the two functions ki ∈C1(Γi; R+

), i = 0, 1, is not identically zero. Let

(∂w

∂ν+ k0w

) ∣∣∣∣Σ0

T

≡ 0,

(∂w

∂ν+ k1w

) ∣∣∣∣Σ1

T

≡ 0.

In addition, if there does not exist a function k ∈ C1(Γ; R+) such thatk∣∣Γ0

= k0 and k∣∣Γ1

= k1 then, we require (as in B.C.-2) junction J (defined

in 15) to fall within the dissipative region, away from ΓU : J ⊂ int(ΓC\ΓU ).

Assumption 5 (Geometry for the case of unobserved Neumann b.c.). When theunobserved boundary houses Neumann-type dynamics, i.e. ΓU 6= ∅ and either B.C.-2 with (16), or B.C.-3, then ΓU must satisfy stronger assumptions with convexitybeing a sufficient condition. More generally we can assume:

(a) Let the “unobserved” surface ΓU be given as a level set

ΓU = x ∈ Rn : ℓ(x) = 0, ℓ ∈ C4, ∇ℓ 6= 0 on ΓU (18)

where map ℓ is defined on a suitable domain in Rn.(b) The Hessian matrix of ℓ is non-negative definite on ΓU :

Hℓ

∣∣ΓU

≥ 0 (19)

which is a characterization for the surface z = ℓ(x) to be convex or havingconvex epigraph, or for the set Ω to be convex near ΓU , so that ℓ(x) ≤ 0 forx ∈ Ω near ΓU , and ∇ℓ points towards the exterior of Ω.

Remark 3. Alternatively in place of (14), (19) can respectively assume (x−x0)·ν ≥0 and Hℓ

∣∣ΓU

≤ 0. See [34, p. 302] for more details.

Remark 4. We do not assume the interface J (15) to be empty, thus when Γ0 andΓ1 are subject to different boundary dynamics, the solutions may develop singu-larities at higher energy levels [21]. A model with J = ∅, for instance when Ω isan annulus, is simpler since one does not need to deal with the loss of regularityinduced by the mixed conditions. We will not explicitly focus on this latter case.

Remark 5. Note that in the mixed cases we require junction J to fall insideΓC \ ΓU . Under Dirichlet Neumann mixed boundary (B.C.-2) this condition isnecessary to guarantee sufficient regularity of strong solutions; this regularity isneeded to perform the energy estimates. In Robin-Neumann case (B.C.-3) theinterior regularity of the flow is not an issue, nevertheless the restriction J ⊂ ΓC\ΓU

is still required to avoid singularities in the traces on the unobserved boundary ΓU .In general, one may consider more complicated mixed conditions consisting of

multiple Dirichlet / Neumann / Robin segments, but the strategy of the proofs willremain the same as long as all singularities on the boundary fall inside the closureof the dissipative subdomain.

A two-dimensional section of a suitable domain under mixed b.c. can be viewedon Figure 1.

LONG-TERM DYNAMICS OF A WAVE EQUATION 465

Figure 1. Example of a 2-dimensional cross-section of Ω for thecase of mixed boundary conditions.

Hypothesis (14) is a classical tools employed to study long-term stability un-der restricted damping. This assumption guarantees the necessary geometric op-tics condition [6] in the case of localized boundary dissipation. The presence ofa Neumann-type segment combined with the absence of dissipation around it de-mands more structural regularity from the boundary, with (14), (18), (19) beingsufficient assumptions [34, 35]. The intrinsic reason for this complication is thata Neumann system does not satisfy Lopatinskii condition and the sharp trace reg-ularity result (the so-called “hidden regularity”) enjoyed by Dirichlet models, nolonger applies. Therefore, to obtain information on the unobserved traces one hasto account for geometrical properties of the domain announced in Assumptions 4and 5. For this purpose the next sections we introduce special cutoff functions ζ, ξand a smooth vector field h.

1.4.1. Smooth cutoffs ξ, ζ. Because Γ is compact, while each of the sets Γ, ΓU , ΓC

is connected and relatively open in Γ (possibly with ΓU = ∅), we can construct twofunctions ξ, ζ ∈ C2(Ω; [0, 1]) with the following properties:

1. Map ζ accounts for the data from the damped region:

supp (ζ) ⊂ Ωχ (20)

2. Map ξ does not pick up information from Γ \ ΓU , i.e.: ξ(Γ \ ΓU ) = 03. For every point x in the domain at least one of ξ(x), ζ(x) equals 1:

x ∈ Ω : ξ(x) < 1 ⊂ ζ−1(1) (21)

Remark 6. Assumption (21) is more convenient for our purposes than justdemanding ξ, ζ to form a partition of unity. The overlap of the unity regionswill make it easier to deal with the commutators of ∆ with Mξ and Mζ insubsequent energy estimates.

466 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

So ξ and ζ approximately partition Ω into dissipative and non-dissipative sub-domains. Now define z := ξz. Multiply each term in (8) by ξ, then using [[·, ·]] toindicate a commutator we write

ztt − ∆z + [[∆, Mξ]]z + ξχg(zt) = ξf(z) in QT (22)

where Mξ denotes pointwise (a.e.) multiplication by ξ. Similarly define z := ζz and

ztt − ∆z + [[∆, Mζ ]]z + ζχg(zt) = ζf(z) in QT . (23)

The partitioning of the boundary by segments ΓU , ΓC and the supports of ξ, ζare illustrated on Figure 2.

Figure 2. Schematic covering of Γ by ΓU , ΓC and by the sup-ports of cutoff maps ξ, ζ. This diagram represents the case ofunobserved Neumann-type boundary. Under mixed boundary con-ditions the junction J (15) must fall into the dissipative collar:J ⊂ int(ΓC \ ΓU ). Under full homogeneous Dirichlet condition, orRobin boundary with a C1(Γ) - coefficient, location of J becomesirrelevant.

1.4.2. Vector field h. The shape of the segment ΓU (when nonempty) “reflects” theenergy towards the damped region Ωχ. This property of ΓU can be captured in aspecial vector field h which helps to recover the traces on the unobserved boundary.Consider the following cases depending on which b.c. applies (B.C.-1, 2, or 3) :

I. The simplest situation arises when ΓU = ∅, then pick any x0 ∈ Rn. Whereas,if ΓU 6= ∅, and the unobserved boundary is of Dirichlet type: either B.C-1, orB.C.-2 with (17), then Assumption 4 holds and we can use x0 as in (14). Nowdefine

d(x) :=1

2|x − x0|

2. (24)

LONG-TERM DYNAMICS OF A WAVE EQUATION 467

II. When the unobserved boundary is of Neumann type, namely ΓU 6= ∅ andeither B.C.-3, or B.C.-2 with (16) applies, we use a different approach. By(14) we have a map d0(x) := 1

2 |x− x0|2, which satisfies Dνd0 = (x− x0) · ν ≤0 on ΓU .

Following the method in [34, p. 302], which was originally inspired byprivate communication with Daniel Tataru, we define (instead of (24))

d(x) := d0(x) −

(∂d0

∂ν

)ℓ

|∇ℓ|+ λℓ2 + κ0 (25)

where ℓ is as given in Assumption 5.

The translation constant κ0 in (25) is needed to ensure

infx∈Ω

d(x) > 0 (26)

we can simply let κ0 be sufficiently large, which does not affect properties of ∇d.Next, using the definition (24) or (25), whichever case applies, let

h(x) := ∇d(x). (27)

The key properties of the vector field h are

• For (25) we have [34, pp. 301 – 303]

∂d

∂ν

∣∣∣∣ΓU

= (h · ν)∣∣ΓU

≡ 0 (28)

• The Hessian matrix Hd of d (same as the Jacobian Jh of h) evaluated on ΓU

is positive definite. In particular we can extend d to some open set containingall of Ω so that for some ρ > 0

Hd = Jh ≥ ρId ∀x ∈ Ω. (29)

This property follows immediately for (24), and we refer the reader to [34, pp.301 – 303] in the case of (25).

1.5. Previous work. Until recently the research of wave dynamics had been pri-marily focused on systems with subcritical nonlinearities, addressing only specialinstances of critical models. Subcritical sources under internal linear dissipationwere originally considered in [3, 20, 23, 25, 45], and the first extension to a criticalsource appeared in [2]. For wave equations with linear interior dissipation see also[4, 39] and references therein. Results of [38] address the nonlinear setup: subcriti-cal source with nonlinear (subcritical) interior damping. Earlier studies of criticalexponents in the context of hyperbolic equations were limited to some specific cases:one-dimensional models [16], or large size damping requirements [10, 15, 24, 27], orstructured nonlinearity [17, 37] where growth of the source was correlated with thegrowth of the damping.

In [11] the authors for the first time addressed the issue of nonlinear interiordissipation and “double-critical” exponents: for both dissipation and the source,verifying existence and finite-dimensionality of a global compact attractor for asemilinear wave equation.

All of the aforementioned works deal with full interior damping. The study oflocalized damping began with models subject to boundary dissipation. Propertiesof attractors for the wave equation with linear boundary damping and a subcriticalsource can be found in [33, 28]; the latter reference investigates convergence of themodel with linear damping acting on a collar of the domain, to the model with (full)

468 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

boundary dissipation. Nonlinear boundary damping was studied in [8] showing theexistence of a global compact attractor; the finite-dimensionality result for a similarmodel was later verified in [9], however, for a sub-critical source only. Paper [18]proves the existence of a global attractor in the case of localized nonlinear dampingacting on a full collar of the domain (i.e. ΓC = Γ), and Dirichlet b.c.

Another recent paper [30] deals with critical exponents in a locally damped linearplate equation on an unbounded domain; however, analysis of plates, or more gen-erally Petrowski-type systems, is substantially simpler because the inherent higherorder of the finite energy renders the action of the source on the gradient of displace-ment inessential (i.e. of a lower-order), despite critical exponents. Consequently,for a plate equation the criticality of the source emerges only in conjunction withkinetic energy of the solutions. Such an argument does not hold for the wave equa-tion, where the product of the source with the gradient of the displacement hasfull finite-energy level, and therefore requires a much more refined assessment. Infact, analysis of this critical product via Carleman’s estimates constitutes one ofthe main novel contributions of our paper.

The ultimate purpose of the this article is to explore the issue of critical exponentscombined with localized damping. We aim at verifying the existence and propertiesof the global attractor under restricted interior dissipation and a critical sourceterm.

Finally, let us note that recent years have witnessed a considerable interest inlong-term behavior and global attractors for less traditional PDE that have beenmotivated by physical applications (see [14, 29, 40, 41]). Our model, due to localizeddamping and mixed boundary conditions, falls into a similar category.

2. Well-posedness.

2.1. Abstract framework and energy functionals. We denote by A the pos-itive self-adjoint differential operator −∆ defined on square-integrable functionswhose distribution derivatives of second order reside in L2(Ω):

A = −∆ on L2(Ω)

D(A) = w ∈ L2(Ω) : ∆w ∈ L2(Ω), and w satisfies (B.C.-1, 2, or 3) .

To study of Hadamard’s well-posedness of the problem (1) we cast it into a moreabstract setting. Write Y (t) for the transpose of the vector w(t), wt(t). System(1) can then be reformulated as a Cauchy problem

Yt + AY + BY = 0 Y (0) = w0, w1 (30)

on the finite energy space the finite energy space

H := D(A1/2) × L2(Ω). (31)

Here

A :=

[0 −IdA χ(x)g

]

B :=

[0 0−f 0

]

D(A) =u, v ∈ H : v ∈ D(A1/2), Au ∈ L2(Ω)

, D(B) = H .

Remark 7. More generally, for an arbitrary continuous monotone damping g (withg(0) = 0) one can describe the domain of the generator as: D(A) =

u, v ∈

H : v ∈ D(A1/2), Au + χ(x)g(v) ∈ L2(Ω).

LONG-TERM DYNAMICS OF A WAVE EQUATION 469

With system (1) we associate the following energy functionals. Given any w ∈C1([0, T ]; D(A1/2)) we define the quadratic energy of its state at time t to be thesquare of the norm in the phase-space:

Ew(t) :=1

2‖w(t), wt(t)‖

2H =

1

2‖wt(t)‖

2 +1

2‖A1/2w(t)‖2. (32)

Recalling that F ′(s) = f(s) we also introduce the full energy:

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

∫

Ω

F (w(t))dΩ. (33)

2.2. Existence, uniqueness and basic energy relations: Theorem 2.3. First,let us recall the notion of a solution to the nonlinear problem (1) (e.g. see [5]):

Definition 2.1 (Strong solution). An a.e. defined function w(x, t) is said to be astrong solution of the boundary value problem (1) with (B.C.-1, 2, or 3) on theinterval [0, T [ if

i) w ∈ W 1,∞(0, T ; D(A1/2)

)∩ W 2,∞(0, T ; L2(Ω))

ii) The pair w, wt satisfies equation (1) in the following sense:

d+

dtwt(t) + Aw(t) + χg

(d+

dtw(t)

)

= f(w(t)) a.e. t ∈ [0, T [

w(0) = w0, wt(0) = w1

where the right-derivative (d+/dt)wt is considered in the topology of L2(Ω) and(d+/dt)w in D(A1/2).

Definition 2.2 (Generalized (weak) solution).A function w ∈ C1

(]0, T [; D(A1/2)

)∩ C1

(]0, T [; L2(Ω)

)is a generalized solution

to the boundary value problem (1) with (B.C.-1, 2, or 3) on the interval [0, T [ ifthere exists a sequence of strong solutions wn(t)∞n=1 with initial data wn

0 , wn1

∞n=1

such that

limn→∞

∥∥wn, wn

t − w, wt∥∥

C([0,T [,H )= 0.

Theorem 2.3. Suppose Assumptions 1 and 2 hold.

I. [Existence, uniqueness] If w0, w1 ∈ H , then there exists a unique (non-linear) semi-flow S(t) : w0, w1 7→ w(t), wt(t) on H so that w(t) is ageneralized solution to (1) with (B.C.-1, 2, or 3) on the interval [0, T [ forany T < ∞. If, in addition, w0, w1 ∈ D(A), then w(t) is a strong solutionof (30).

The solutions satisfy variational equality of the following form

(wtt, ϕ) + (A1/2w, A1/2ϕ) + 〈χg(wt), ϕ〉 = (f(w), ϕ) (34)

for all t ∈ [0, T [ and ϕ ∈ D(A1/2). Here 〈·, ·〉 stands for the duality pair-ing [D(A1/2)]′ × D(A1/2). For generalized trajectories, the product (wtt, ϕ) =ddt (wt, ϕ) should be interpreted as the distributional derivative.

II. [Regularity] Suppose w0, w1 ∈ D(A), then the corresponding strong so-lution remains bounded in the domain of the principal evolution generator:w, wt ∈ L∞(0, T ; D(A)) for all T < ∞. Moreover:(a) Let U(J) denote some Rn-neighborhood of the junction J (15). Then for

any T < ∞

w, wt ∈ L∞(0, T ; H2(Ω∗) × H1(Ω)

)

470 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

where Ω∗ is the interior of Ω\U(J). When J = ∅ (e.g. if Ω is an annulus),one can replace Ω∗ by Ω.

(b) If Neumann-Robin B.C.-3 holds, then

w, wt ∈ L∞(0, T ; H2−η(Ω) × H1(Ω)

)

for all T < ∞. When there exists a function k ∈ C(Γ) which coincideswith k0 on Γ0 and with k1 on Γ1 one can choose η = 0.

III. [Lipschitz property] The semi-flow S(t) is locally Lipschitz on H :

‖S(t)Y1 − S(t)Y2‖H ≤ eb(r)‖Y1 − Y2‖H , ∀‖Yi‖H ≤ r, t ≥ 0. (35)

IV. [Fundamental energy relations] Suppose w, wt is a generalized solu-tion defined on the interval [t0, t1[, then for any and [s, t] ⊂ [t0, t1[

Ew(t) +

∫ t

s

D(wt) = Ew(s) +

∫ t

s

(f(w), wt). (36)

The difference of trajectories z = w − u satisfies an analogous equation:

Ez(t) +

∫ t

s

D(zt) = Ez(s) +

∫ t

s

(f(z), zt). (37)

Note that (36) can be rewritten (using the definition (33) of E(t)) morenaturally as

Ew(t) +

∫ t

s

D(wt) = Ew(s). (38)

In addition, there exist positive constants c0, c1, dependent only on f andΩ, so that

c0Ew(t) − c1 ≤ Ew(t) ≤ Ew(0), ∀t ≥ 0. (39)

V. [Gradient system] The functional t 7→ Ew(t) is a strict Lyapunov functionfor dynamical system H , S(t). Consequently, by definition (e.g. see [12]),H , S(t) is a gradient system.

2.3. Proving local existence of solutions to (1). The local result [0, Tmax[some Tmax < ∞ follows from the theory of monotone operators on Banach spaces.The maximal monotonicity property of a nonlinear operator of the form A is shownin [5, pp. 302-304]. To apply the result in our case it suffices to observe that the sets, g(s)s∈R is a maximal monotone graph in R ×R. Hence the Nemytski operatorassociated to g, is the sub-differential of the convex function ϕ : D(A1/2) →]−∞,∞]defined via

ϕ(y) =

∫

Ω

χ(x) j(y(x))dΩ with j : x 7→

∫ x

0

g(s)ds.

Consequently, the nonlinear generator A + B of the evolution (30) is a locally Lip-schitz perturbation of a maximal monotone operator. Existence and uniqueness ofsolutions on some time interval [0, Tmax[ have been shown in [8, Theorem 7.2].

2.4. Regularity and Lipschitz character of the flow. Within the interval ofexistence, on [t0, t1] ⊂ [0, Tmax[, locally Lipschitz perturbation f produces the sameevolution trajectory if we replace the Nemytski operator corresponding to the sourceterm with an appropriately chosen globally Lipschitz map (see the proof of [8, The-orem 7.2]). Therefore all the properties of the flow S(t) stem from the theory ofmonotone generators with globally Lipschitz perturbations. Thus, by appealing to

LONG-TERM DYNAMICS OF A WAVE EQUATION 471

the result [5, Theorem 1.1 (Ch. 4)] we may claim that trajectories of the semi-flowS(t) continuously depend on initial condition, which confirms (35).

The spatial regularity of strong solutions follows from the fact that initial datafrom the domain of the monotone generator (with a Lipschitz perturbation) yieldssolutions that remain in that domain (e.g. see [43, Theorem 4.1 Ch. IV]). So theregularity at time t coincides with the domain of the “principal” generator A. Thenthe velocity component wt(t) resides in D(A1/2) ⊂ H1(Ω), and Aw(t)+χ(x)g(wt) ∈L2(Ω). From the linear bounds on g at infinity we conclude that g(wt) ∈ L2(Ω)and w(t) ∈ D(A).

A more precise characterization of D(A) depends on the shape of the boundaryΓ and the b.c. In a regular situation (no change of b.c.), D(A) falls into H2(Ω).Classical elliptic theory states that in mixed problems when J 6= ∅ we can guaranteethis regularity only on subsets disjoint from some neighborhood of the interface J .

In a smooth domain, global (in space) regularity of the w for mixed Dirichlet-Neumann condition B.C.-2 is H3/2−η(Ω) [42] (which might be improved only if thedomain has a crease of a certain aperture at the junction [22]). The Neumann-Robinsituation is more favorable: here the regularity of w goes up to almost two weakderivatives: namely w(t) ∈ H2−η(Ω), the proof is given in [35, p. 1784].

The uniformity in time: L∞(0, T ) for T < Tmax, is likewise implied by theregularity of solutions to systems with smooth initial data and a globally Lipschitzsource: see for instance [5, Theorem 1.4 & Equation 1.1 (Ch. 4)], with possibleblowup as T ր Tmax. We will be able to replace Tmax by ∞ if we prove thatsolutions are globally bounded.

2.5. Proof of the energy identity. Suppose initial data is smooth, then we canuse strong variational formulation (34) with the test function wt. Furthermore, Sec-tion 2.4 shows that we have sufficient regularity to integrate the strong variationalformulation in time on the interval [s, t]. Then the classical argument of integrationby parts (in space and time) confirms (36).

Because g is linearly bounded at infinity all terms of (36) are continuous withrespect to the finite-energy topology. Consequently by approximating the weaksolutions with strong ones (converging in C([s, t]; H )) we verify the energy identityfor generalized trajectories.

2.6. Proving global existence of solutions to (1). Recall that local existenceup to time Tmax ultimately followed from [8, Theorem 7.2]. The latter result alsostates that if Tmax < ∞ then limtրTmax

‖w(t), wt(t)‖H = ∞. Thus, to replaceTmax with any T < ∞ we will need a priori bounds on the norm of the solutionin the phase space, in particular it is sufficient to show (39), which follows fromthe argument in [8, p. 1913]. For the sake of completeness we place this proof inAppendix (Proposition a-9). As a consequence the all the statements so far can beextended to time-interval [0, T ] for any T < ∞.

2.7. Proof of the gradient system property. We have to show that E(t) is astrict Lyapunov function.

Identity (38) implies that the map t 7→ E(w(t), wt(t)) is non-increasing in t forstrong solutions, and by continuity of Ew in the finite energy topology we concludethe same for weak solutions.

Furthermore, suppose we are given a generalized solution w, wt, so that Ew(t) =Ew(0) for any t ∈ R+. Pick any approximating sequence of strong trajectories

472 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

wn, wnt convergent to w, wt, then

∫

QTχg(wn

t )wnt → 0 implying that wn

t → 0

pointwise a.e. in Ωχ. Since the sequence wnt converges to wt in C([0, T ]; L2(Ω)) we

deduce wt

∣∣Ωχ×[0,T ]

≡ 0 for T < ∞. The unique continuation result in [46] (which is

a combination of: the argument in [19], and a slight modification of the uniquenessresult in [34]) implies that wt ≡ 0 for all t > 0, i.e. w, wt = w,0 is a stationarysolution of (1).

The two aforementioned properties verify that E is a strict Lyapunov functionfor the dynamical system (H , S(t)). This section completes the proof of Theorem2.3.

3. The main results. We recall (e.g. see [3], [7], or [45]) the following

Definition 3.1 (Global attractor). A bounded closed set A ⊂ H is said to be aglobal 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

limt→+∞

dist H (S(t)B, A ) = 0.

To describe the attractor, we introduce the set N of stationary points of theflow S(t):

N := W ∈ H : S(t)W = W ∀t ≥ 0. (40)

Every stationary point W ∈ N has the form W = (w, 0) and solves the ellipticboundary value problem

−∆w = f(w) in Ω; w satisfies B.C.-1, 2, or 3.

Condition (10) on f readily implies that the set N of equilibria is bounded (seeProposition a-10 in the Appendix).

In order to precisely describe the structure attractor we will need a few moredefinitions.

Definition 3.2 (Unstable manifold of stationary points). Suppose there exists acurve in the phase-space t 7→ Y (t), 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 suchY (t) a full trajectory.

Define the unstable manifold Mu(N ) emanating from N as the set of all fulltrajectories Y (t) which satisfy

limt→−∞

dist H (Y (t), N ) = 0.

Definition 3.3 (Fractal dimension, e.g. see [7]). The fractal dimension dimf M ofa compact set M can be defined by the formula

dimf M := lim supr→0+

lnN(M , r)

ln(1/r)(41)

where N(M , r) is the minimal number of closed sets of diameter 2r which coverM .

The main results read:

LONG-TERM DYNAMICS OF A WAVE EQUATION 473

Theorem 3.4 (Global compact attractor). Suppose Assumptions 1, 2 apply. Inaddition, adopt Assumption 4 (when ΓU 6= ∅), and Assumption 5 (when ΓU 6= ∅and either B.C.-2 with (16), or B.C.-3). Then the dynamical system H , S(t)possesses a global, compact attractor A ⊂ H which coincides with the unstablemanifold of stationary points:

(i) A = Mu(N )(ii) lim

t→+∞dist H (S(t)W, N ) = 0 for any W ∈ H .

Remark 8. Existence of a global compact attractor for a wave equation withlocalized differentiable damping: g ∈ C1 and Dirichlet b.c. has been shown in[18]. Paper [8] proves this result for a wave equations with differentiable boundarydamping. Differentiability of g is used critically in both [18] and [8] and the methodsemployed there do not generalize to the non-smooth case. Instead, the result statedin Theorem 3.4 does not require g to be differentiable.

Theorem 3.5 (Finite-dimensionality and regularity). Let the assumptions of The-orem 3.4 apply, and, in addition, adopt a stronger condition on g: Assumption3. Then the fractal dimension of the global attractor A of the dynamical systemH , S(t) is finite. Furthermore, A is bounded in D(A), in particular A is abounded subset of H2(Ω) × H1(Ω).

Remark 9. To our knowledge this is the first result which proves finite-dimensionality and smoothness of the attractor for a hyperbolic flow with local-ized dissipation and a critical source.

4. Proofs of the main Theorems. Existence and properties of the attractorclaimed in Theorems 3.4 and 3.5 ultimately follow from abstract results on dissi-pative dynamical systems, once we establish appropriate inverse-type observabilityinequalities. The latter constitute the main contribution of this paper and arepresented in this section, while all the technical details of their proofs have beenrelegated to Sections 5 and 6.

4.1. Existence and structure of the attractor: Theorem 3.4. Existence andthe structure of the attractor are closely related to the asymptotic smoothness ofthe semi-flow S(t).

Definition 4.1. A dynamical system X, S(t) is said to be asymptotically smooth(e.g. see [23]) if for any bounded set B ⊂ X that is forward invariant (i.e. S(t)B ⊂B for t ≥ 0), there exists a compact set K ⊂ B such that

limt→+∞

supY ∈B

dist XS(t)Y, K = 0.

Let us recall a recent criterion for asymptotic smoothness due to [31, Thm. 2]. Wepresent it in the abstract formulation used in [11] and [12] which is more convenientfor applications.

Proposition 1 (Asymptotic smoothness [11, 12]). Let (X , S(t)) be a dynamicalsystem on a complete metric space X with a metric d. Assume that for any boundedpositively invariant set B ⊂ X and any ε > 0 there exists T = T (ε, B) such that

d(S(T )Y1, S(T )Y2

)≤ ε + Ψε,B,T (Y1, Y2) ∀Yi ∈ B

474 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

where Ψε,B,T (Y1, Y2) is a functional on B × B such that for every sequenceYn∞n=1 ⊂ B

lim infm→∞

lim infn→∞

Ψε,B,T (Yn, Ym) = 0. (42)

Then X , S(t) is an asymptotically smooth dynamical system.

The estimate in Proposition 1 follows from the following lemma:

Lemma 4.2 (First Inverse-Observability Inequality). Let B denote a bounded sub-set of H ; pick Y1, Y2 ∈ B with Y1 = w0, w1 and Y2 = u0, u1. Define z = w−uto be the difference of corresponding trajectories. Let cutoff map ξ and vector fieldh be as defined respectively in Sections 1.4.1 and 1.4.2, and set z = ξz.

Under the assumptions of Theorem 3.4 we have:

1. If solutions w and u are strong, then for any sufficiently large T > 0 and anyε, ε > 0, there are positive constants C, Cε, Cε,ε, CT,ε for which the followingidentity holds:

(C − ε)

∫ T

0

Ez(t)dt ≤Cε

∫

QT,ε

(max g(±ε)2 + ε2)

+ Cε,ε

∫

QT \QT,ε

χ(x)(g(wt)wt + g(ut)ut

)

+

∫

QT

ξf(z)h · ∇zdQ + C(Ez(0) + Ez(T ))

+ CT,ε‖z‖2C([0,T ];L2(Ω))

(43)

where

QT,ε ≡(x, t) ∈ Ωχ × [0, T ] : |ut(x, t)| + |vt(x, t)| < ε

2. As a corollary of estimate (43), even if w and u are generalized solutionsoriginating in B, there exists a constant CB,ε > 0 satisfying

Ez(T ) ≤ C(max g(±ε)2 + ε2) +CB,ε

T+ ΨB,T (Y1, Y2) (44)

where

ΨB,T (Y1, Y2) := CT supt∈[0,T ]

‖z(t)‖2 + CT

∫ T

0

ds

∫ T

s

(f(z), zt) + CT

∫

QT

ξ2f(z)h · ∇z.

The proof of Lemma 4.2 occupies Section 5. Proposition a-11 in the Appendixshows that functional Ψ possesses required compensated compactness property (42)and, therefore, verifies the hypothesis of Proposition 1. Since, in addition, (H , S(t))is a gradient system (Theorem 2.3 part V), and the set N of equilibria is bounded(Proposition a-10 in the Appendix), there exists a global compact attractor whichcoincides with the unstable manifold Mu(N ) [12]. This argument completes theproof of Theorem 3.4.

4.2. Dimension and regularity of the attractor: Theorem 3.5. The boundon fractal dimension of the attractor A stems from the following abstract result(e.g. see [10, Theorem 2.10], or [12, Theorem 2.15]).

Proposition 2 (Finite dimensionality [12, Theorem 2.15]). Let X be a Banachspace and M be a bounded closed set in X .

LONG-TERM DYNAMICS OF A WAVE EQUATION 475

Assume one can find a Lipschitz map V : M → X such that M ⊂ V M , andthere exist compact seminorms n1(x), n2(x) on X satisfying

‖V v1 − V v2‖X ≤ σ‖v1 − v2‖ + K(n1(v1 − v2) + n2(V v1 − V v2)) (45)

for any v1, v2 ∈ M , where 0 < σ < 1 and K > 0 are constants. (A seminormn(x) on X is said to be compact if for any bounded set B ⊂ X there exists asequence xm ⊂ B such that n(xm − xn) → 0 as n, m → ∞; here the factor K, ofcourse, does not affect compactness, however, K is related to the estimates on thedimension [12], which is why we chose to leave this parameter in the statement).

THEN, M is a compact set in X of a finite fractal dimension.

The task of exhibiting property (45) ultimately reduces to showing that the dif-ference of any two trajectories through the attractor can be stabilized exponentiallyto a compact set. The key is the following Carleman-type inequality:

Lemma 4.3 (Second Inverse-Observability Inequality (Carleman-type)). Adopt hy-potheses of Theorem 3.4 (in particular, g only needs to satisfy Assumption 2). LetA be the global compact attractor of the system (H , S(t)), whose existence wasverified it Theorem 3.4. Suppose that w0, w1, u0, u1 ∈ A with z = w − u.

Recall definitions of the smooth cutoff maps ξ, ζ in Section 1.4.1, and vector fieldh in Section 1.4.2. Let z = ξz and z = ζz. Then

1. For a sufficiently large T > 0, any positive parameters τ, C (eventually large), andε, ε0 (eventually small), there exist positive constants C1, C2, CT , CC,T , Cτ,T,ε,ε0,C

so that

C1

ENERGY OVER supp ξ︷ ︸︸ ︷∫

QT

eτΦ(|∇(ξz)|2 + (ξzt)

2)+C

ENERGY OVER supp ζ︷ ︸︸ ︷∫

QT

eτΦ(|∇(ζz)|2 + (ζzt)

2)−

−CT

∫ T

0

∫

ζ−1(1)

eτΦ(|∇z|2 + z2

t

)

≤ − τ

∫

QT

e−τΦM21 +

∫

QT

ξ(

f(z) − χg(zt))

M1

+ C2

∫

QT

eτΦχ2g(zt)2 + 2C

∫

Ωχ

∫ T

0

eτΦz2t

ε0

4

∫ T

0

‖∇z‖2 + ε

∫

QT

eτΦ(|∇z|2 + z2

t

)dQ

+ CC,T e−δτ [Ez(T ) + Ez(0)]

+ Cτ,T,ε,ε0,C‖z‖2C([0,T ];L2(Ω))

(46)where Φ(x, t) is a certain smooth pseudo-convex function (defined later in (59))and

M1 = eτΦ[h · ∇(ξz) − Φtξzt]

2. If g also satisfies Assumption 3, and the initial condition belongs to the attractorA , then, as a corollary of (46), there exist positive constants σ < 1 and CB,σ > 0such that

Ez(s + T ) ≤ σEz(s + T ) + CB,σ supθ∈[0,T ]

‖z(s + θ)‖2 (47)

476 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

for a sufficiently large T > 0 (as in (46), any starting time s ∈ R

The proof of Lemma 4.3 is presented in Section 6.

Remark 10. The structure of inequality (46) gives some indication on how toderive (47) from it: the first two terms on LHS will be shown to dominate the totalquadratic (free) energy, while C will be chosen large enough to absorb the thirdintegral on the LHS. On the right, a sufficiently large value of parameter τ will helpto dispense with the influence of the source: f(z)M1; in addition, τ will be used

to handle Ez(T ) an Ez(0). The kinetic term C∫

Ωχ

∫ T

0eτΦz2

t can be related to the

damping χg(zt) because the former gathers information only over the support ofcutoff χ(x) – precisely where the dissipation is active.

To apply Lemma 4.3 we proceed as in [10] (see also [38]). Define

X := H × W (0, T ) (48)

where

W (0, T ) :=

w ∈ H1(]0, T [ ; L2(Ω)

): ‖w‖W := ‖w, wt‖L2(0,T ;H ) < ∞

in particular: 12

∥∥w(0), wt(0), w(t)

∥∥

2

X= Ew(0) +

∫ T

0Ew(t)dt. Now introduce a

special subset of X :

AT :=

Y0, S(t)Y0

∣∣[0,T ]

: Y0 ∈ A

. (49)

Consider the “shift” operator V = VT : AT → X

V : Y0, S(t)Y0 7→ S(T )Y0, S(T + t)Y0. (50)

We will show that the map V satisfies the conditions of Proposition 2. Result (35)states that the semi-flow S(t) is locally Lipschitz on H , hence V is locally Lipschitzon (the bounded set) AT . In addition V AT = AT .

Integrate (47) over all s ∈ [0, T ]:∫ 2T

T

Ez(t) ≤ σ

∫ T

0

Ez(s)ds + CT supθ∈[0,2T ]

‖z(θ)‖2

adding the result to the original equation (47) yields

Ez(T ) +

∫ 2T

T

Ez(t) ≤ σ

(

Ez(0) +

∫ T

0

Ez(s)ds

)

+ 2CT supθ∈[0,2T ]

‖z(θ)‖2.

This relation can be recast (after extracting square roots) into the form

‖V W1 − V W2‖X ≤ σ‖W1 − W2‖X + KT · (nT (W1 − W2) + nT (V W1 − V W2))

for any W1, W2 ∈ AT , and

nT (W ) = supθ∈[0,T ]

‖w(θ)‖2.

The seminorm nT is compact on X (see the proof of Proposition a-11 in the Ap-pendix, especially (a-120)). Since σ < 1, we can apply Proposition 2 to claim thatAT is a compact set in X of finite fractal dimension.

Let P : X → H : Y0, S(t)Y0 7→ Y0, then projection P is Lipschitz continuousand A = PAT . The definition (41) implies that fractal dimension of a set is no

LONG-TERM DYNAMICS OF A WAVE EQUATION 477

more than the dimension of its pre-image under a Lipschitz map. Thus dimH

f (A ) ≤

dimX

f (AT ) < ∞.The regularity result stating that A is bounded in the domain of the evolution

generator, is another consequence of (47) as shown in Proposition 7 (see the end ofSection 6.8.3). This argument completes the proof of Theorem 3.5.

The remainder of the paper is devoted to the proofs of Lemmas 4.2 and 4.3.

5. Proving First Inverse-Observability Inequality (Lemma 4.2). Unlessotherwise stated all solutions are strong. Later we will extend the results to in-clude the generalized trajectories as well.

5.1. Fundamental identity. To verify inequalities stated in Lemma 4.2 we pro-ceed in two steps: (i) establish an extended energy identity (Proposition 3 below);(ii) estimate each term in the identity depending on which of the four formal cat-egories this term belongs to: damping, energy, source, or lower-order (Proposition4).

Let cutoff maps ξ, ζ be as defined in Section 1.4.1, and vector field h be given by(27). Recall that z = ξz, z = ζz.

Proposition 3 (Fundamental identity). Suppose hypotheses of Theorem 3.4 hold,and initial conditions are smooth: w0, w1, u0, u1 ∈ D(A), z = w − u. Recallcutoffs ξ, ζ from Section 1.4.1 and vector field h from Section 1.4.2. Let z = ζz,z = ξz. Then the following identity holds:

(BΣ) =

∫

QT

(Jh − ρId)∇z · ∇z +ρ

2

∫

QT

(|∇z|2 + z2t )

+ C

∫

QT

(|∇z|2 + z2t ) − 2C

∫

QT

z2t +

1

2

∫

QT

z∇z · ∇div h

+

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z))[

h · ∇z +z

2(div h − ρ)

]

+ C

∫

QT

(

ζχg(zt) + [[∆, Mζ ]]z − ζf(z))

z

+ k0,T

(51)

where

(BΣ) :=

∫

ΣT

∂z

∂νh · ∇z +

1

2

∫

ΣT

∂z

∂νz(div h − ρ) +

1

2

∫

ΣT

(z2

t − |∇z|2)(h · ν)

+ C

∫

ΣT

(∂z

∂ν

)

z

and

k0,T =

[∫

Ω

zth · ∇z

]T

0

+1

2

[∫

Ω

ztz (div h − ρ)

]T

0

+ C

[∫

Ω

ztz

]T

0

.

Proof. To obtain (51) one can employ either of the two arguments:

(a) Use the fundamental identity derived in [46] (in particular, add the equations(43) and (54) from [46]).

478 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

(b) Alternatively, one can arrive at (51) as a special case of Proposition 5 givenbelow. Simply set τ = 0, Φt = 0, and c = 0 in (64). The discarded termsare not crucial because the machinery of Carleman estimates is not required toshow asymptotic smoothness of the flow.

End of the proof of Proposition 3.

Let us give a brief overview of equation (51):

• Integral∫

QT(Jh − ρId)∇z · ∇z has a definite sign due to the strict positivity

of Jh, while the terms∫

QT(|∇z|2 + z2

t ),∫

QT(|∇z|2 + z2

t ) are equivalent to the

energy∫ T

0 E(t) which is the quantity we want to estimate.

• Product (ξf (z), h∇z) presents the key issue, because it is equivalent to theenergy level ‖∇z‖2 (since f is critical). In the subcritical scenario this quantitycould have been estimated by ε‖∇z‖2 plus some lower order terms (below thetopology of the phase space H ), however once the order of the polynomialbound on f reaches the critical Sobolev exponent for the embedding H1 → L2,an entirely different approach has to be used.

A similar challenge will come up from the product (f(z), zt) which is “hid-den” inside the damping terms, but will eventually reappear when we invokethe basic energy identity (37).

Due to relative weakness of the conditions necessary for asymptotic smooth-ness (Proposition 1) one can go around the criticality of the products

(ξf(z), h∇z), and (f(z), zt). However this issue will require a completelydifferent treatment when verifying the stronger result on dimension of theattractor (Lemma 4.3).

• The quantity C∫

QT(|∇z|2 + z2

t ) will be used to absorb the term [[∆, Mξ]]h ·∇z

when C is sufficiently large. The counterpart term −2C∫

QTz2

t is “benign”

because its integrand is supported on the region where the dissipation is active.• Depending on the boundary conditions the traces in (BΣ) either vanish or

produce at most lower-order quantities.

5.2. Bounds on the energy terms. Next we transform fundamental identity(51) into an inequality

Proposition 4 (Intermediate observability inequality). Adopt hypotheses of The-orem 3.4 and let the initial data be smooth w0, w1, u0, u1 ∈ D(A), z = w − u.Then

(Cρ − ε)

∫ T

0

Ez(t)dt ≤−

∫

QT

ξχg(zt)

[

h · ∇z +z

2(div h − ρ)

]

− C

∫

QT

ζχg(zt)z + 2C

∫

QT

z2t

+

∫

QT

ξf(z)h · ∇z + CT,C,ε‖z‖2C([0,T ];L2(Ω))

+ CC(Ez(0) + Ez(T )).

(52)

Proof. Inequality (52) follows once we estimate the terms in (51) exactly as it wasdone in [46]. Alternatively one may refer to Section 6 (in particular subsection 6.5),where the same estimates are presented in detail for a more general case when theidentity includes Carleman weights.

LONG-TERM DYNAMICS OF A WAVE EQUATION 479

The remaining task now is to assess the influence of the terms involving g. Picksome ε > 0; because g is continuous and monotone increasing, the linear bounds atinfinity in (11) make it possible to find, mε, Mε so that

mε|s| ≤ |g(s)| ≤ Mε|s| for |s| ≥ ε. (53)

As a consequence |s| ≥ ε implies g(s)2 + s2 ≤ (Mε + m−1ε )g(s)s. Pick any a.e.

defined version of ut, vt and let

QT,ε ≡(x, t) ∈ Ωχ × [0, T ] : |ut(x, t)| + |vt(x, t)| < ε

.

We have

−

∫

QT

ξχg(zt)

[

h · ∇z +z

2(div h − ρ)

]

− C

∫

QT

ζχg(zt)z + 2C

∫

QT

z2t

≤CC,ε

∫

QT,ε

(max g(±ε)2 + ε2) + CC,ε,ε

∫

QT \QT,ε

χ(x)(g(wt)wt + g(ut)ut

)

+ ε

∫

QT

|∇z|2 + Cε‖z‖2C([0,T ];L2(Ω)).

(54)

Apply inequality (54) to rewrite the RHS of (52), which proves the first estimate(43) of Lemma 4.2.

Next, we are going to verify the rest of Lemma 4.2. Observe that (38) and

(39) provide us with an L1( [0,∞[ ) bound on the dissipation product:∫ T

0 D(wt) ≤Ew(0) + c1 (same holds for ut). As a result

CC,ε,ε

∫

QT \QT,ε

χ(x)(g(wt)wt + g(ut)ut

)≤ CC,ε,ε,B.

Next, expand the term containing the source (this is just to express ∇(z) in termsof ∇z and ∇ξ):∫

QT

ξf(z)h · ∇z =

∫

QT

ξ2h · f(z)∇z +

∫

QT

ξh · (∇ξ)f(z)z

≤

∫

QT

ξ2h · f(z)∇z + ε

∫

QT

|∇z|2 + CT,ε‖z‖2C([0,T ];L2(Ω)).

(55)

Fix t = T in the basic energy identity (37) and integrate over all s ∈ [0, T ]:

TEz(T ) ≤

∫ T

0

E(s)ds +

∫ T

0

ds

∫ T

s

(f(z), zt). (56)

Apply (54, 55) to (52), move ε∫|∇z|2 to LHS, combine with (56), and finally divide

both sides by T (observe that |QT,ε|/T ≤ const = |Ωχ| regardless of the value ofε):

Ez(T ) ≤ C(max g(±ε)2 + ε2) +CB,ε

T+ ΨB,T (Y1, Y2) (57)

where Y1 = w0, w1, Y2 = u0, u1 and

ΨB,T (Y1, Y2) := CT supt∈[0,T ]

‖z(t)‖2 + CT

∫ T

0

ds

∫ T

s

(f(z), zt) + CT

∫

QT

ξ2f(z)h · ∇z.

(58)Note that each term in inequality (57) is continuous with respect to the finite energytopology H . Hence by density the same relation also holds for all generalized

480 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

solutions; this statement completes the proof of the second estimate (44) in Lemma4.2. Proof of Lemma 4.2 is now complete.

By first selecting sufficiently small ε, and then by increasing T we can make theleading terms C(max g(±ε)2 + ε2)+CB,ε/T in (57) as small as we wish. Therefore,according to Proposition 1, in order to verify asymptotic smoothness of the systemH , S(t) we only have to show that ΨB,T satisfies the limit condition (42). Theproof of the latter statement is similar to the one in [11], and we relegate theargument to the Appendix (Proposition a-11). Thus, Lemma 4.2 and Propositiona-11 verify the hypotheses of Proposition 1, which, in turn, implies asymptoticsmoothness of system (H , S) (i.e. the statement of Theorem 3.4).

6. Proving Second Inverse-Observability Inequality (Lemma 4.3).

6.1. Overview. The ultimate purpose of this section is to prove inequality (47).This goal is attained in several steps: (i) study the action of the strong variationalformulation (34) on specially constructed test-functions, which will eventually leadus to Proposition 5; then (ii) estimate the terms involved in that identity to get abound on quadratic energy: Proposition 6; (iii) finally, estimate interaction of thesource with the velocity component (these energy the estimates will take advantageof the structure of the attractor): Propositions 7 and 8.

Generally speaking, all the derivations are aimed at bounding Ez(T ) from aboveby the dissipation, lower order norms of z (norms in topologies below the energylevel H ), and by initial data (however, the coefficient of initial energy must beindependent of time T ). Due to relative equivalence of kinetic and potential ener-gies it will be sufficient, for the most part, to estimate just the kinetic component‖zt(T )‖2 alone. In the region where the damping is active one can directly relatethe velocity to damping χ(x)g(zt)zt, whereas over the unobserved subdomain weexpress the energy in terms of the boundary traces (on ΓU ) which are dispensedwith via the geometrical assumptions on the unobserved boundary. Thus, in orderto carry out the analysis we will split the domain into two parts: the damped, andthe unobserved regions, by respective cutoff maps ξ and ζ.

Below is a formal schematic outline of the computations: it indicates how theenergy over Ω is expressed in terms of the dissipation, and lower order terms (l.o.t.– norms below the finite energy level H ); in addition one will eventually have todeal with the contribution of the source.

Ω \ Ωχ −→ΓU

Dirichlet: (h · ν) ≤ 0Neumann: (h · ν) = 0

≤ l.o.t.

րEnergy on Ω +[source]

ց

Ωχ

(Energy ∼ kinetic)−→ ≤

T∫

0

D(zt) + l.o.t.

6.1.1. Critical growth of the source feedback map. Some aspects of our approachare similar to the ones in [46]; however, the main obstacle now stems from the factthat we deal with a difference of two trajectories. The key issue here lies in ourinability to estimate (by standard means) the RHS perturbation f(z) = f(w)−f(u)when f is critical. Classical weighted multiplier techniques introduce the following

LONG-TERM DYNAMICS OF A WAVE EQUATION 481

quantities:∫

QT

f(z)h · ∇z and

∫

QT

f(z)zt.

Because the polynomial bound on f carries a critical exponent, then ‖f(z)‖ ∼‖z‖1,Ω, so the above two products are equivalent to the energy level ‖∇z‖ ‖zt‖ andmay distort the estimates because we have no direct control over their sign andmagnitude.

Remark 11. If f were sub-critical (e.g. |f(s)| ≤ |s|3−ε in 3 dimensions) the abovequantities could have been readily estimated as ε(‖∇z‖ + ‖zt‖) plus norms of z intopologies below H1. The sub-critical scenario is significantly easier and requires amuch shorter argument; most of the subsequent derivations are designed preciselyto handle the challenging borderline case |f(s)| ∼ |s|3.

When examining a single trajectory, one can take advantage of the identitiesf(w)∇w = ∇F (w), f(w)wt = ∂tF (w), which eventually render the perturbationf(w) benign. The proof of asymptotic smoothness (Lemma 4.2), in particularProposition a-11 in the Appendix, took advantage of this approach because theiterated limits in (42) permitted to analyze one trajectory at a time. However, this

method no longer works if we directly compare Ez(t) to (f(z), h · ∇z) or (f(z), zt),which is what we must do to establish finite-dimensionality.

To handle the product (f(z), h ·∇z) we use Carleman’s estimates, while the term

(f(z), zt) can be assessed if we exploit the structure and compactness of the globalattractor (which were established in Theorem 3.4).

6.2. Regularity requirements. Until stated otherwise we will be exclusively work-ing with strong solutions to the original equation (1) and its difference version (8).The regularity of strong trajectories guaranteed by Theorem 2.3 is necessary tojustify the computations below. More specifically, we need higher regularity of so-lutions on the part of the domain not covered by the damping. The calculus in theproof requires at least z ∈ H3/2+η(Ω). When mixed conditions B.C.-2 are applied,the junction J falls outside the support of z, whence we do not pick up any singu-larities propagating from the boundary and z ∈ H2(Ω). On the damped portionΩχ the multipliers are of a lower order and it suffices to have z ∈ H1(Ω) to carryout the proof.

In case of mixed Robin-Neumann condition B.C.-3 with a discontinuity betweenk0 and k1, we can still claim sufficient regularity z ∈ H2−η(Ω) regardless of thelocation of J . Nevertheless, we still demand J to fall outside the support of z inorder to estimate the traces on the unobserved boundary.

6.3. Pseudo-convex function Φ(x, t). So far we have shown the construction ofspecial smooth cutoffs ξ, ζ (Section 1.4.1) and vector field h (Section 1.4.2). Car-leman estimates will, in addition, involve a certain pseudo-convex function (whosegradient coincides with h).

Recalling d(x) from Section 1.4.2, and let Φ : Ω × R → R be defined by

Φ(x, t) = Φ(x, t; T ) := d(x) − c

∣∣∣∣t −

T

2

∣∣∣∣

2

(59)

where constants

T > 0 and 0 < c < min

1,ρ

2

(60)

482 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

are selected so that

cT > 2 maxy∈Ω

√

|d(y)|. (61)

The last condition on T is a consequence of the finite speed of energy propagation,it ensures that T is large enough to observe the entire system from dissipativesubdomain.

Remark 12. For another reference on multipliers based on pseudo-convex functionssee [26].

We immediately have

Φ(x, 0) = Φ(x, T ) = d(x) − cT 2

4< |d(x)| − c2 T 2

4< 0

Hence there is a constant δ > 0

Φ(x, 0) < −δ < 0; Φ(x, T ) < −δ < 0 for all x ∈ Ω (62)

Consequently, according to (26) there exists an interval [t0, t1] ⊂]0, T [ such that

Φ(x, t) > 0, for t ∈ [t0, t1], x ∈ Ω. (63)

Remark 13. In fact it suffices to have Φ(x, t) > −δ for t ∈ [t0, t1]. This relationwill be used only once later on in inequality (92), and there exponent −δτ could bereplaced by −(δ + min[t0,t1] Φ(x, t))τ which is still positive, as long as Φ(x, t) > −δon the interval. But since condition (26) is a property that does not require anyadditional assumptions on the model, we can always arrange it so that (63) holds.

Observe, that ∇Φ(x) = h(x). Henceforth, we will be using ∇Φ and h inter-changeably.

6.4. Fundamental-identity: Carleman-version (Proposition 5). The firststep in the proof of Lemma 4.3 is the following equation:

Proposition 5 (Fundamental identity (Carleman-type)). Suppose the hypotheses ofTheorem 2.3 hold. Choose pseudo-convex function Φ(x, t) = Φ(x, t; T ) as describedin Section 6.3, and let h = ∇Φ. Let ξ, ζ be as in Section 1.4.1, and recall thenotation z = ξz, z = ζz. Then for any value of parameter τ > 0, and any positiveconstant C we have:

(∫

QT

eτΦ(Jh − ρId)∇z · ∇z

)

+ [Energy]ξ+C[Energy]ζ

=[Damping]− τ

∫

QT

e−τΦM21

+

∫

QT

ξ(

f(z) − χg(zt))

M1

−

∫

QT

([[∆, Mξ]]z)M1

+ [Almost lower order]

+ (BΣ)τ − k0,T − Ck0,T

(64)Where

M1 = eτΦ[h · ∇(z) − Φtzt]

LONG-TERM DYNAMICS OF A WAVE EQUATION 483

µ = div (eτΦh) − ∂t(eτΦΦt) (65)

[Energy]ξ := (ρ/2 − c)

∫

QT

eτΦ(|∇z|2 + z2t ) ; [Energy]ζ :=

∫

QT

eτΦ(|∇z|2 + z2t )

[Almost lower order] := −

∫

QT

z∇z · ∇(µ

2− (ρ/2 + c)eτΦ

)

+

∫

QT

z ztd

dt

(µ

2− (ρ/2 + c)eτΦ

)

+C

∫

QT

zztd

dt(eτΦ) − C

∫

QT

z∇z · (∇eτΦ)

−

∫

QT

(

[[∆, Mξ]]z − ξf(z)) [

z(µ

2− (ρ/2 + c)eτΦ

)]

−C

∫

QT

(

[[∆, Mζ]]z − ζf(z))

eτΦz

[Damping] := 2C

∫

QT

eτΦz2t −

∫

QT

ξχg(zt)z(µ

2− (ρ/2 + c)eτΦ

)

−C

∫

QT

ζχg(zt)eτΦz

(BΣ)τ =

∫

ΣT

∂z

∂νM1 +

∫

ΣT

∂z

∂νz(µ

2− (ρ/2 + c)eτΦ

)

+1

2

∫

ΣT

eτΦ(z2

t − |∇z|2)(h · ν) + C

∫

ΣT

(∂z

∂ν

)

zeτΦ

k0,T =

[∫

Ω

eτΦzt(h · ∇z)

]T

0

−1

2

[∫

Ω

eτΦΦt

(z2

t + |∇z|2)]T

0

+

[∫

Ω

ztz(µ

2− (ρ/2 + c)eτΦ

)]T

0

k0,T =

[∫

Ω

ztzeτΦ

]T

0

.

Remark 14. The term∫

QTeτΦz2

t is listed as “damping,” because z2t is supported

within suppχ, which makes it possible to relate z2t to the dissipation g(zt). Also

note: though the construction of Ψ requests T to be large enough (in (61)), thiscondition is not necessary for this proposition to hold (for now, we can simply ignore(61) and pick any positive constants c and T ).

The identity in Proposition 5 is self-contained and does not make use of theprecise structure of Φ, h, and the cutoff maps ξ, ζ. This relation can be derived forwave equation (1) independently of the present discussion, provided strong solutionshave sufficient regularity. For this reason we move the lengthy and rather technicalproof to the Appendix (Section 7.3).

6.5. The bounds on the total energy (Proposition 6). Our goal is to obtainan upper bound on the [Energy]-type terms in fundamental identity (64).

484 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

Proposition 6 (Intermediate energy estimate). Suppose the hypotheses of Theorem3.5 are satisfied. Let w0, w1, u0, u1 belong to some bounded set B ⊂ H (notnecessarily smooth, i.e. the corresponding solutions w and u could be generalizedtrajectories). As before, let z = w−u be the difference of the two trajectories. Thenfor all sufficiently large T > 0, there exists 0 < σ < 1 and positive constants CT ;1,CT ;2, CB,T such that

Ez(T ) + CT ;1

∫ T

0

Ez(t)dt ≤ σEz(0) + CB,T supt∈[0,T ]

‖z(t)‖2 + CT ;2 [Source terms]

(66)

where

[Source terms] =

∫ T

0

dt

∣∣∣∣

∫ t

0

(f(z), zt)

∣∣∣∣+

∫ T

0

ds

∣∣∣∣∣

∫ T

s

(f(z), zt)

∣∣∣∣∣+

∣∣∣∣∣

∫ T

0

(f(z), zt)

∣∣∣∣∣.

(67)

This proposition will eventually follow from fundamental identity (64). Below isthe outline of the estimates that we will perform on (64):

• The influence of the source ξfM1 is handled with the help of the term−τ∫

QTe−τΦM2

1 once τ is chosen to be sufficiently large.

• Strict positivity Jh ≥ ρId implies that the RHS dominates [Energy]-terms(while the left-most integral can be discarded from the resulting inequality).

• The term ([[∆, Mξ]]z)M1 has full energy level, however is only supportedwithin the region where the damping is active (due to the overlap of the cutoffsξ and ζ). This quantity will be absorbed into C[Energy]ζ for a sufficientlylarge C.

• All damping terms will be estimated via χ(x)g(zt)zt which will eventuallybe rewritten using the basic energy identity (37). This procedure will bring

out another energy-level product f(z)zt that will be dealt with using theknowledge on the structure of the attractor (as established by Theorem 3.4).

6.6. Establishing Part 1 of Lemma 4.3. The next set of estimates in the proofof Proposition 6 will also help us establish the intermediate inequality (46) of Lemma4.3.

Positivity of the Jacobian Jh

¿From (29) we have

0 ≤

∫

QT

eτΦ(Jh − ρId)∇z · ∇z. (68)

Consequently this integral can be discarded if we change the equal sign in (64) to“≤.”

Using of Carleman weights to absorb the critical source termAs was mentioned before, the key difficulty of managing a difference of two

trajectories lies in our inability to gauge the term f(z)h · ∇z, which appears inside

f(z)M1. This is where we exploit the magnitude of (large) parameter τ (see alsoRemark 19 in the Appendix).

LONG-TERM DYNAMICS OF A WAVE EQUATION 485

For any ε > 0 we can write

−τ

∫

QT

e−τΦM21+

∫

QT

ξ(

f(z) − χg(zt))

M1

≤−

(

τ −1

2ε

)∫

QT

e−τΦM21 + ε

∫

QT

eτΦ(f(z)2 + χ2g(zt)2).

(69)

Remark 15. Note that in (69) we used parameter τ to transform χg(zt)M1 intoa small multiple of g(zt)

2. In fact, we will not be able to use the magnitude of ε tohandle g(zt)

2, and instead, later on, will have to invoke condition (12). However,such a mollification of the damping term might prove useful in other contexts, whichis why we indicate the possibility of carrying it out.

To relate f and Ez(t) we will also need the following inequality

eτΦ/2|f(z)| ≤eτΦ/2

∫ 1

0

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

whence via Holder’s inequality and Sobolev imbedding H1(Ω) → L6(Ω), we have

‖eτΦf(z)2‖0,1,Ω ≤ C∥∥1 + w4 + u4

∥∥

0,3/2,Ω

∥∥eτΦz2‖0,3,Ω ≤ CB‖eτΦ/2z‖2

1,Ω

≤ CB‖eτΦ/2∇z‖2 + CB,T,τ‖z‖2.

(70)

Interior terms of a lower orderLet us first relate the potential energy with its pieces: ∇z and ∇z. Note that

∇ξ vanishes whenever ξ attains (its maximum) value 1; the same applies to ζ.Consequently, (21) along with the fact that |∇ξ|, |∇ζ| are uniformly bounded onΩ, gives a pointwise estimate on |∇z| + |∇z|:

|∇z| ≤

(

|∇z| + |∇z|

)

≤ Cξ,ζ (|∇z| + |z|) (71)

Since Φ and h are smooth functions on QT , we have another useful inequality:(before placing the bound, expand each term to factorize eτΦ)

|∇µ| + |∇eτΦ| + |∂tµ| + |∂teτΦ| ≤ Cτ,T eτΦ ∀t, x ∈ QT .

Note also [[∆, Mζ ]] = (∆ζ)z + 2∇ζ∇z (and similarly for Mξ). Now repeatedly useab ≤ (εc)a2/2 + b2/(2cε) to estimate the lower order interior terms in fundamentalidentity (64). This procedure may be carried out in several different ways, the onlyguide is that we always place a factor of ε next to the (energy-level) integrands

eτΦz2t , eτΦ|∇z|2 and eτΦf(z)2.

[Almost lower order] ≤ε

∫

QT

eτΦ(

|∇z|2 + z2t + f(z)2

)

dQ

+ Cτ,T,C,ε ‖z‖2C([0,T ];L2(Ω))

(72)

We did not track the structure of the constant in front of ‖z‖2C([0,T ];L2(Ω)) because

the L2(Ω)-norm of the z falls below the finite energy level.

486 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

The dissipation termsThe next inequality deals with some of the terms on the RHS of (64) that involve

damping g. Using the fact that z is supported on Ωχ derive

[Damping] ≤ 2C

∫

Ωχ

∫ T

0

eτΦz2t +

∫

QT

eτΦχ2g(zt)2 + Cτ,T,C‖z‖

2C([0,T ];L2(Ω)). (73)

The cutoff commutatorsIntroduction of the cutoffs ξ, ζ has brought commutators into the estimates; in

particular, we should be concerned with the product ([[∆, Mξ]]z)(h · ∇z), that isequivalent to quadratic energy term |∇z|2. However, this quantity appears benignas its support falls within the dissipative region, where we can take advantage ofthe damping. Let

Ωζ = ζ−1(1) ∩ Ω. (74)

According to (20), Ωζ ⊂ Ωχ, furthermore, due to the property (21) we have x :ξ(x) < 1 ⊂ x : ζ(x) = 1, whence [[∆, Mξ]]

∣∣Ω\Ωζ

≡ 0. Recall the definition (65)

of µ and that |Φt(x, t)| ≤ cT for t ∈ [0, T ].

∫ T

0

∫

Ωζ

([[∆, Mξ]]z)

M1︷ ︸︸ ︷

eτΦ (h · ∇z − Φtzt) ≤

∫ T

0

∫

Ωζ

Ch,ξTeτΦ(|∇z|2 + z2

t

)

+ Cτ,T ‖z‖2C([0,T ];L2(Ω))

(75)

For some constant Ch,ξ dependent only on h and ξ.The boundary terms – Case: ΓU = ∅

The goal of this subsection is to arrive at the inequality (81) stated below, in thecase ΓU = ∅. Here we have

(BΣ)τ = C

∫

ΣT

∂z

∂νz

the latter term vanishes in case of full Dirichlet or mixed Dirichlet-Neumann homo-geneous conditions, whereas under Robin boundary we rewrite the normal compo-nent in terms of lower order quantities and directly pass to inequality (81) via thecontinuous trace map H1−η(Ω) → L2(Γ) for any 0 ≤ η < 1/2.

The boundary terms – Case: ΓU is of Dirichlet typeWhen ΓU is nonempty and subjected to Dirichlet dynamics, we may discard the

tangential component, and use relation h = ∇d = (x− x0) along with the property(14) to derive:

∫

ΣT

eτΦ ∂z

∂ν(h · ∇z) +

1

2

∫

ΣT

eτΦ(z2t − |∇z|2)(h · ν) =

1

2

∫

ΣT

eτΦ

(∂z

∂ν

)2

(h · ν) ≤ 0.

The remaining terms in (BΣ)τ are estimated exactly as the integral in the CaseΓU = ∅ above, thus leading us to (81).

The boundary terms – Case: ΓU is of Neumann type

LONG-TERM DYNAMICS OF A WAVE EQUATION 487

Finally, when a Neumann-type condition acts on ΓU , use Assumption 5 whichprovides the identity h · ν = 0 on ΓU (see (28)). Therefore,

(BΣ)τ =

∫

ΣT

∂z

∂ν

M1︷ ︸︸ ︷

eτΦ(h · ∇z − Φt · zt)+

∫

ΣT

∂z

∂νz

[1

2µ − (ρ/2 + c)eτΦ

]

+ C

∫

ΣT

(∂z

∂ν

)

zeτΦ.

(76)

With the exception of the first integral, all the terms can be estimated exactly asthe one in the Case ΓU = ∅ above.

The remaining integrand ∂z∂νM1 disappears under mixed Dirichlet-Neumann (or

full Dirichlet) homogeneous boundary conditions. Therefore we only need to dealwith the Robin setting (B.C.-3) when k0 and k1 are restrictions of the same non-negative C1 function k on Γ (when such a function k does not exist, the singularityon the junction J falls outside the support of z, hence the argument will be thesame when k is replaced by the appropriate ki). Start with

∫

ΣT

∂z

∂νeτΦh · ∇z = −

∫

ΣT

eτΦkzh · ∇z = −

∫

ΣT

eτΦkzh ·

(∂z

∂νν + ∇tanz

)

=

∫

ΣT

eτΦk2z2(h · ν) −

∫

ΣT

eτΦkzh · ∇tanz

(77)

where ∇tanz =∑n−1

j=1 (∇z · αj)αj , and x 7→ aj(x)n−1j=1 is a smooth frame on

manifold Γ; operator div tan will indicate the divergence on Γ.Because Γ has no boundary, we obtain∫

ΣT

eτΦkzh · ∇tanz =1

2

∫

ΣT

eτΦkh · ∇tan(z2)

=1

2

∫

ΣT

div tan

(eτΦkhz2

)−

1

2

∫

ΣT

z2div tan

(eτΦkh

)

= −1

2

∫

ΣT

z2div tan

(eτΦkh

).

(78)

Use (78) to rewrite the RHS of (77); then (with H1−η(Ω) → L2(Γ) for 0 ≤ η < 1/2)we get the inequality

∫

ΣT

∂z

∂νeτΦh · ∇z ≤ Cτ,T

∫ T

0

‖z‖21−η,Ω. (79)

The task of estimating the term∫

ΣT

∂z∂ν eτΦΦt inside

∫

ΣT

∂z∂νM1 is simpler:

∫

ΣT

∂z

∂νeτΦΦt = −

∫

ΣT

eτΦΦtkzzt = −1

2

∫

ΣT

kd

dt

(eτΦΦtz

2)

+1

2

∫

ΣT

kz2 d

dt

(eτΦΦt

)

≤ Cτ,T

∫ T

0

‖z‖21−η,Ω + Cτ,T ‖z‖

2C([0,T ];L2(Ω)).

(80)

Inequalities (79) and (80) provide an estimate on∫

ΣT

∂z∂νM1. Let ε0 > 0 be a (po-

tentially very small) parameter whose value we will specify later. Putting together

488 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

the above bounds on the summands in (BΣ)τ , we arrive at

(BΣ)τ ≤Cτ,T,C

∫ T

0

‖z(t)‖21−η,Ω + Cτ,T,C‖z‖C([0,T ]; L2(Ω))

≤ε0

4

∫ T

0

‖∇z‖2 + Cτ,T,C,ε0‖z‖2

C([0,T ];L2(Ω))

(81)

where in the last step we used an interpolation estimate to rewrite ‖z‖1−η alongwith (71).

Remark 16. In (81), the factor (ε0/4), can be changed to any value below ε0/2.

The terminal products (t = 0, T )

Finally we need to deal with the quantities evaluated at the end-points of thetime-interval. Using (61) and (62) obtain

|k0,T + Ck0,T | ≤ CCTe−δτ [Ez(T ) + Ez(0)] + Cτ,T,C‖z‖2C([0,T ];L2(Ω)). (82)

Applying the estimates to the fundamental identity (64)

Use estimates (68), (69), (72), (73), (75), (81), and (82) to transform fundamentalidentity (64) of Proposition 5 into the following inequality

(ρ/2 − c)

∫

QT

eτΦ(|∇z|2 + z2t )+C

∫

QT

eτΦ(|∇z|2 + z2t )−

−

∫ T

0

∫

Ωζ

Ch,ξTeτΦ(|∇z|2 + z2

t

)

≤ε0

4

∫ T

0

‖∇z‖2 + 2ε

∫

QT

eτΦ(

|∇z|2 + z2t + f(z)2

)

dQ

−

(

τ −1

2ε

)∫

QT

e−τΦM21

+ C

∫

QT

eτΦχ2g(zt)2 + 2C

∫

Ωχ

∫ T

0

eτΦz2t

+ CCTe−δτ [Ez(T ) + Ez(0)]

+ Cτ,T,C,ε,ε0‖z‖2

C([0,T ];L2(Ω)).

(83)

If in the previous step we had not rewritten terms involving τ via (69), theninstead of (83) we would have obtained the first part (46) of Lemma 4.3.

The next few sections will complete the proof of Proposition 6, which in turn willbe used to verify the second statement of Lemma 4.3.

6.7. Choosing values of the free parameters τ, C (large), and ε (small).We shall further simplify (83) by adjusting the free parameters τ , ε, and C:

I. Choose

ε = τ−1 (84)

in which case(

τ −1

2ε

)∫

QT

e−τΦM21 ≥ 0. (85)

LONG-TERM DYNAMICS OF A WAVE EQUATION 489

II. Since Ωζ (see (74)) falls into the pre-image of 1 under map ζ, we get

(C − Ch,ξT )

∫

QT

eτΦ(|∇z|2 + z2t ) ≤C

∫

QT

eτΦ(|∇z|2 + z2t )

−

∫ T

0

∫

Ωζ

Ch,ξTeτΦ(|∇z|2 + z2

t

).

(86)

Because |h|, ξ are uniformly bounded on Ω, we may choose C to satisfy (recallthat ρ > 2c by (60))

C = CT > Ch,ξT + ρ/2 − c. (87)

III. Use (70) to derive the following(instead of dependence on ε we will only indicate dependence on τ since wehave set ε = τ−1)

2ε

∫

QT

eτΦf(z)2 ≤ εCB

∫

QT

eτΦ|∇z|2 + CB,τ,T ‖z‖C([0,T ];L2(Ω)). (88)

IV. An application of (71) and (87) gives

(ρ/2 − c)

∫

QT

eτΦ(|∇z|2 + z2t )+(C − Ch,ξT )

∫

QT

eτΦ(|∇z|2 + z2t )

−2ε

∫

QT

eτΦ(|∇z|2 + z2t ) − εCB

∫

QT

eτΦ|∇z|2

≥ (ρ/4 − c/2 − 2ε − εCB)

∫

QT

eτΦ(|∇z|2 + z2t )

= Cρ

∫

QT

eτΦ(|∇z|2 + z2t ).

(89)

Since ε = τ−1 and ρ/2 > c, the leading factor Cρ on the RHS of (89) can bemade positive provided τ > (2 + CB)(ρ/4 − c/2)−1.

V. Use assumption (12) on the damping to conclude g(zt)2+z2

t ≤ (M+m−1)g(zt)zt.Then

2C

∫

Ωχ

∫ T

0

eτΦz2t + C

∫

QT

eτΦχ2g(zt)2 ≤CC

∫

QT

eτΦχg(zt)ztdQ

≤Cτ,T,C

∫ T

0

D(zt)dt.

(90)

VI. Finally, we must separate the energy terms z2t , |∇z|2 from the Carleman weight

eτΦ. Use condition (63) to derive

Cρ

∫ t1

t0

∫

Ω

(|∇z|2 + z2t )dt ≤ Cρ

∫

QT

eτΦ(|∇z|2 + z2t ).

The norms ‖A1/2z‖ and ‖∇z‖ are equivalent: to prove it start with the identity(Az, z) =

∫

Ω |∇z|2−∫

Γ∂z∂ν z, then note that the product ∂z

∂ν z is either 0 or equal

to (const · z2), depending on the boundary conditions. Consequently∫

Γ∂z∂ν z

can be bounded by C‖z‖21−η,Ω some 0 < η < 1/2. Now estimate ‖z‖1−η,Ω by

490 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

‖z‖1,Ω and apply Poincare’s inequality. In particular we end up with

Cρ,Ω

∫ t1

t0

Ez(t)dt ≤ Cρ

∫

QT

eτΦ(|∇z|2 + z2t ). (91)

Now sequentially apply (85), (86), (88), (89), (90), and (91) to inequality (83).Rearrange the summands, divide both sides by Cρ,Ω:

∫ t1

t0

Ez(t)dt ≤Cτ,T

∫ T

0

D(zt)dt + CT e−δτ [Ez(T ) + Ez(0)]

+ε0

4

∫ T

0

‖∇z‖2 + Cτ,T,ε0‖z(t)‖2

C([0,T ];L2(Ω))

(92)

(We no longer need to track the dependence on C.)

Energy near the terminal pointsInequality (92) loses track of the energy over [0, t0]∪ [t1, T ] which must be some-

how recovered, since the damping term on the RHS gathers data from [0, T ], and

that (implicitly) carries the issue of the source (f(z), zt) being integrated over entireinterval [0, T ].

The only other tool at our disposal is the basic energy identity (37). Set s = 0in (37) and integrate the latter over all t ∈ [0, T ]. After dropping the non-negativedissipation integral on the left, obtain

∫ T

0

Ez(t)dt ≤ TEz(0) +

∫ T

0

dt

∫ t

0

(f(z), zt). (93)

The drawback of this inequality is the potentially large factor in front of Ez(0).So we first multiply (93) by ε0 add only then the result to (92). The integral

(ε0/4)∫ T

0 ‖∇z‖2 from the latter inequality can be estimated by (ε0/2)∫ T

0 Ez(t) andabsorbed into the LHS. Consequently

∫ t1

t0

Ez(t)dt +ε0

2

∫ T

0

Ez(t)dt ≤Cτ,T

∫ T

0

D(zt)dt

+ CT e−δτ (Ez(T ) + Ez(0)) + ε0TEz(0)

+ Cτ,T,ε0‖z(t)‖2

C([0,T ];L2(Ω))

+ ε0

∫ T

0

dt

∫ t

0

(f(z), zt).

(94)

Remark 17. Alternatively we could have derived a pointwise (in time) boundon Ez(T ) from the energy identity (37) by using Gronwall’s inequality, and thenintegrated the result to get an estimate similar to (93).

Recovering E(T ) pointwise

To obtain pointwise (in time) energy from the LHS of (94) set t = T in (37) andintegrate it over s ∈ ]t0, t1[.

(t1 − t0)Ez(T ) ≤

∫ t1

t0

Ez(s)ds +

∫ t1

t0

ds

∫ T

s

(f(z), zt) (95)

LONG-TERM DYNAMICS OF A WAVE EQUATION 491

Multiply equation (94) by 2 and add the result to (95). Cancel one instance of∫ t1

t0Ez(s)ds on each side. Then

(t1 − t0)Ez(T ) + ε0

∫ T

0

Ez(t)dt ≤ 2Cτ,T

∫ T

0

D(zt)dt

+ 2CT e−δτ (Ez(T ) + Ez(0)) + 2ε0TEz(0)

+ 2Cτ,T,ε0‖z‖2

C([0,T ]; L2(Ω))

+ 2ε0

∫ T

0

dt

∫ t

0

(f(z), zt) +

∫ t1

t0

ds

∫ T

s

(f(z), zt).

(96)

Expand the dissipation integral∫ T

0D(zt)dt via the energy relation (37), and carry

all terms containing Ez(T ) to the LHS:(

2Cτ,T − 2CT e−δτ + t1 − t0

)

Ez(T )+ε0

∫ T

0

Ez(t)

≤

(

2Cτ,T + 2CT e−δτ + 2ε0T

)

Ez(0)

+ 2Cτ,T‖z‖2C([0,T ]; L2(Ω))

+ 2ε0

∫ T

0

dt

∫ t

0

(f(z), zt)

+ C

∫ t1

t0

ds

∫ T

s

(f(z), zt)

+ CτT

∫ T

0

(f(z), zt).

(97)

The crux of the matter is that for a sufficiently large τ and small ε0 we have

(t1 − t0) − 2CT e−δτ > 2CT e−δτ + 2ε0T.

For this choice of parameters the coefficient of Ez(T ) on the left of (97) exceedsthe coefficient of Ez(0) on the RHS. Consequently, if normalize the factor in frontof Ez(T ), then E(0) in the previous estimate will acquire a factor 0 < σ < 1.Afterwards, use a trivial upper bound on all the integrals involving f to acquire thedesired inequality (66).

To complete the proof of Proposition 6 it remains to show that (66) holds forweak solutions as well. Note that (66) is continuous with respect to the finite energytopology of H . Thus, by approximating generalized solutions with strong ones(converging in C([0, T ]; H ) -topology), we can extend the result to all trajectoriesin the phase space. We have proved Proposition 6.

6.8. Final estimates (proving Part 2 of Lemma 4.3, and the regularityof attractor A ). In this section we will finish the proof of the inequality (47) inLemma 4.3, and will verify smoothness of the attractor (which will complete theproof of Theorem 3.5). These results ultimately follow from (66) once we estimatethe source-dependent terms. In particular we must show that the critical (energy-

level) product (f(z), zt) can be expressed via lower order norms, plus an arbitrarilysmall energy-level perturbation.

492 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

Estimate (66) is similar to the one obtained in [11], however the proof of finite-dimensionality in the latter cannot be readily adapted to our case due to geomet-rically restricted damping. In particular: we could (following [11]) estimate kineticcomponent z2

t by the dissipation g(zt)zt; however, afterwards the argument breaksdown due to the following: L1(R+)-integrability of χg(zt)zt (implied by the energyidentity (36)) does not extend to L2(R+, L2(Ω))-integrability of velocity zt, sincedissipation χg(zt) is comparable to the velocity only on the subset of the domaindetermined by the support of χ.

Instead, we employ the method which was originally introduced in [32] to studyvon Karman equation with internal damping, and then later used for boundary-damped wave (see [11]) and von Karman (see [13]) equations. See also [12] forthe abstract realization of this idea. The approach is based on compactness of theattracting set. Overall the argument first establishes that every trajectory throughthe attractor is a strong trajectory, and then invokes the compactness of A in orderto refine (66). In the end one concludes that the attractor not only consists ofstrong trajectories, but is also uniformly bounded in strong topology.

6.8.1. Intermediate regularity result. In this section we prove an auxiliary proposi-tion, which later on will be invoked twice: first time to prove that the combination(f(z), zt) of the source with kinetic energy is almost of a lower order (Proposition8); and the second time to finally confirm that attractor is uniformly bounded inhigher-order topology (which is the last part of Theorem 3.5).

Proposition 7 (Conditional regularity result). Assume the following:

(a) That hypotheses of Theorem 3.5 have been satisfied.(b) Assume that for a given trajectory u(t), ut(t) through attractor A , for any

constant h ∈]0, 1[, and any ε > 0 there exists

T εu = T (u; ε) > 0

possibly dependent on u and ε, but independent of h; so that for s < t ≤ T εu the

difference of trajectories zh(t) = u(t + h) − u(t), satisfies∣∣∣∣

∫ t

s

(f(zh), zht )

∣∣∣∣≤ ε(Ezh(s) + Ezh(t)) + CA ,ε sup

θ∈[s,t]

‖zh(θ)‖2 + ε

∫ t

s

‖∇zh‖2 (98)

where CA ,ε > 0 is independent of u and h.

THEN for any large T > 0, and sufficiently small ε > 0

‖utt(t)‖2 + ‖ut(t)‖

21,Ω ≤ CA , ∀t < T ε

u − T (99)

where CA depends only on the diameter of (compact) attractor A in the finite-energy topology of H .

Proof. Since u(t) passes through the attractor, it extends to a full trajectory definedfor all t ∈ R. let w(t) = u(t + h) (i.e. now zh = w − u). Apply propositionProposition 6 to zh with initial time s (instead of 0); then invoke assertion (98) inorder to estimate quantity CT [Source terms] in the resulting version of (66):

CT [Source terms] ≤εCT

[

(T + 1)Ehz (s) + (T + 1)Eh

z (s + T ) + 2

∫ s+T

s

Ehz (t)dt

]

+ CA ,ε,T supθ∈[0,T ]

‖zh(s + θ)‖2 + εCT

∫ T

s

‖∇zh‖2

(∀ s ≤ T εu − T ) .

LONG-TERM DYNAMICS OF A WAVE EQUATION 493

After using the bound ‖∇zh(t)‖ ≤ 2Ehz (t) and relabeling the constants, we may

write

CT [Source terms] ≤εCT

[Eh

z (s) + Ehz (s + T )

]+ CA ,ε,T sup

θ∈[0,T ]

‖zh(s + θ)‖2

+ εCT

∫ T

s

Ehz (t).

Finally, to help us with notation call the contractive factor of Proposition 6, “σ”(instead of “σ”). In the end obtain

Ezh(s + T ) + CT ;1

∫ s+T

s

Ezh(t)dt ≤ σEzh(s) + CA ,T,ε supθ∈[0,T ]

‖zh(s + θ)‖2

+εCT [Ezh(s) + Ezh(s + T )]

+εCT

∫ s+T

s

Ez(t)dt

for s ≤ T εu −T . Substitute ε1 = εCT , then move ε1Ezh(s + T )+ ε1

∫ s+T

s ‖∇zh‖2 tothe LHS of the inequality, and divide by (1 − ε1):

Ezh(s + T ) +CT ;1 − ε1

1 − ε1

∫ s+T

s

Ezh(t)dt ≤(σ + ε1)

1 − ε1Ezh(s)

+ CA ,T,ε1sup

θ∈[0,T ]

‖zh(s + θ)‖2.

Now simply pick ε1 sufficiently small to ensure

σ :=(σ + ε1)

1 − ε1< 1 and

CT ;1 − ε1

1 − ε1> 0.

Hence:

Ezh(s + T ) ≤ σEzh(s) + CA ,T supθ∈[0,T ]

‖zh(s + θ)‖2 ∀s ≤ T εu − T

where σ < 1.Now multiply the result by h−2, setting yh = h−1zh:

Eyh(s + T ) ≤ σEyh(s) + CT supθ∈[0,T ]

‖yh(s + θ)‖2. (100)

Since yh → ut in L2(Ω) as h ց 0, the term ‖yh(t + θ)‖2 is uniformly bounded forsmall h and all θ ≥ 0 by a constant CA , dependent only on the norm of the initialdata (i.e. a fixed multiple of the diameter of A in the finite-energy topology of H ).Hence we can write

Eyh(s + T ) ≤ σEyh(s) + CA ∀s ≤ T εu − T.

Take the supremum over all s ∈]−∞, T εu −T ] on the RHS, then supremum on over

all s ∈] −∞, T εu − 2T ] on the LHS. Because σ < 1 we can now conclude

Eyh(s) ≤ CA ∀s ≤ T εu − T.

Where the RHS is independent of h ∈ ]0, 1[. Passing to the limit h ց 0 obtain thatu, ut belongs to the domain of the differential operator ∂t and

‖utt(s)‖2 + ‖ut(s)‖

21,Ω ≤ CA ∀s ≤ T ε

u − T.

This completes the proof of Proposition 7.

494 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

6.8.2. Handling the criticality of the source at the level of kinetic energy. Now weare ready to show that (f(z), zt) is almost of a lower order whenever z is a differenceof trajectories through the attractor.

Proposition 8 (Decomposition of (f(z), zt)). Adopt hypotheses of Theorem 3.5.Then for any two trajectories w(t), wt(t) and u(t), ut(t) through attractor A

(with z = w − u) and any ε > 0∣∣∣∣

∫ t

s

(f(z), zt)

∣∣∣∣≤ ε(Ez(s) + Ez(t)) + CA ,ε sup

θ∈[s,t]

‖z(θ)‖2 + ε

∫ t

s

‖∇z‖2 (101)

for some constant CA ,ε > 0.

Proof. Step 1: Preliminary decomposition of the source. Begin with∫ t

s

(f(z), zt) =

∫ t

s

∫

Ω

∫ 1

0

f ′(λz + u)z ztdλdQ =1

2

∫ t

s

∫

Ω

∫ 1

0

f ′(λz + u)d

dtz2dλdQ

=1

2

[∫

Ω

∫ 1

0

f ′(λz + u)z2dλdΩ

]t

s

−1

2

∫ t

s

∫

Ω

∫ 1

0

f ′′(λz + u)(λzt + ut)z2dλdQ.

(102)

To estimate∫

Ω

∫ 1

0f ′(λz + u)z2dλdΩ use sequentially: (i) the bound (9) on f ′′; (ii)

Holder’s inequality with conjugate exponents 3, 3/2; and (iii) Sobolev’s embeddingH1−η → L3 followed by an interpolation estimate:

∫

Ω

∫ 1

0

f ′(λz + u)z2dλdΩ ≤C

∫

Ω

(1 + |w|2 + |u|2)z2dΩ

≤C∥∥1 + |w| + |u|

∥∥

2

L6(Ω)‖z2‖L3/2(Ω)

≤CA ‖z‖2L3(Ω) ≤ ε‖∇z‖2 + CA ,ε‖z‖

2

(103)

where ε > 0 is arbitrary. For the other integral in (102) apply Holder’s inequalitytwice: first with conjugate exponents 6 and 6/5, then with exponents 10/6 and10/4:∫

Ω

∫ 1

0

f ′′(λz + u)(λzt + ut) z2dλdΩ ≤CA

(∫

Ω

(|wt| + |ut|)6/5z12/5dΩ

)5/6

≤CA (‖wt‖ + ‖ut‖)‖z‖2L6(Ω)

≤CA (‖wt‖ + ‖ut‖)‖∇z‖2

(104)

where in the last step we used the Sobolev’s embeddings to rewrite ‖z‖2L6(Ω) (in

n = 3 dimensions), assuming CA was adjusted accordingly. Substitute the last twoestimates into (102):

∣∣∣∣

∫ t

s

(f(z), zt)

∣∣∣∣≤ε[Ez(s) + Ez(s + T )] + CA ,ε sup

θ∈[s,t]

‖z(θ)‖2

+ CA

∫ t

s

(‖wt‖ + ‖ut‖)‖∇z‖2.

(105)

Remark 18. Note that the right-most integrand in (105) is still “critical” in asense that its magnitude is comparable to quadratic energy Ez(t)

LONG-TERM DYNAMICS OF A WAVE EQUATION 495

Step 2: Smoothness of the flow near stationary points (negative time scale).Pick a trajectory u(t) through the attractor, then from Theorem 3.4 we know thatu(t), ut(t) asymptotically (as t → ±∞) converges to the set of stationary pointsN :

∀Y0 ∈ A limt→±∞

dist (S(t)Y0, N ) = 0.

Let h ∈]0, 1[; since any stationary point N has the form w, 0, we conclude thatfor a given ε > 0 there exists T ε

u (independent of 0 < h < 1, but possibly dependenton the solution u) so that

‖ut(t)‖ + ‖ut(t + h)‖ ≤ ε/CA ∀t ≤ T εu (CA as in (105))

substituting this relation into (105), with w = u(t + h), wt = ut(t + h) we get∣∣∣∣

∫ t

s

(f(z), zt)

∣∣∣∣≤ ε[Ez(s)+Ez(t)]+CA ,ε sup

θ∈[s,t]

‖z(θ)‖2+ε

∫ t

s

‖∇z‖2 ∀s ≤ t ≤ Tu.

This relation is exactly in the form required by hypothesis (b) of Proposition 7;invoking the latter we conclude

‖utt(t)‖2 + ‖ut(t)‖

21,Ω ≤ CA , t ∈ ] −∞, Tu − T ] (106)

(where we no longer need to track the dependence of Tu on ε).Step 3: Forward propagation of regularity. From the original equation (1), result

(106), and bounds on the finite energy of solutions, we conclude that ‖∆u(t)‖ isalso uniformly bounded for t ∈ ] − ∞, Tu − T ], i.e. the trajectory is strong whent falls below Tu − T . By using forward wellposedness of strong solutions stated inTheorem 2.3 we may claim that t 7→ u(t), ut(t) is a strong solution to the originalproblem. Thus any trajectory through the global attractor is a strong trajectory.Consequently the global attractor A resides in the set D(A) ⊂ H2(Ω) × H1(Ω)(whose regularity was established in Theorem 2.3 Part II).

Step 4: Using compactness of the attractor. Since A is compact, it can beapproximated by a finite lattice of points in the phase space H . In addition weknow from Step 3 above, that this lattice is smooth.

More precisely: pick t ∈ R, then a pair of velocity trajectories ut(t), wt(t) throughthe attractor A belongs to a compact set in L2(Ω) consisting of the elements from

D(A1/2) ⊂ H1(Ω). Hence for any ε1 > 0 there exists a finite set υjN(ε)j=1 ⊂ D(A1/2)

such that we can find indices j1, j2 (possibly dependent on ut(t) and wt(t)) satisfying

‖wt(t) − υj1‖ + ‖ut(t) − υj2‖ ≤ ε1.

Moreover,

sup1≤j≤N(ε)

‖υj‖1,Ω ≤ Cε1(107)

Define, for convenience υj1,j2 := υj1 − υj2 . Now we are going to refine the earlierinequality (104). First:

∫

Ω

∫ 1

0

f ′′(λz + u)(λzt + ut) z2dλdΩ

=

∫

Ω

∫ 1

0

f ′′(λz + u)z2[(λzt + ut) − (λυj1,j2 + υj2)

]dλdΩ

+

∫

Ω

∫ 1

0

f ′′(λz + u)z2(λυj1,j2 + υj2)dλdΩ

(108)

496 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

then estimate each summand:∣∣∣∣

∫

Ω

∫ 1

0

f ′′(λz + u)z2[(λzt + ut) − (λυj1,j2 + υj2)

]dλdΩ

∣∣∣∣

≤

∫

Ω

Cf (1 + |z| + |u|)z2(|zt − υj1,j2 | + |ut − υj2 |

)dΩ

≤CA ‖z‖2L6

(‖zt − υj1,j2‖

2 + ‖ut − υj2‖2)≤ ε1CA ‖∇z‖2

we will handle the second integral on the RHS of (108) analogously; however, uni-form bound (107) on the (finite) approximation lattice permits us to assign to theterm (λυj1,j2 + υj2) a Holder exponent strictly above 2, rendering the other factorsubcritical. Consequently:

∣∣∣∣

∫

Ω

∫ 1

0

f ′′(λz + u)z2(λυj1,j2 + υj2)dλdΩ

∣∣∣∣≤Cε1,A ‖z‖2

1−η,Ω

≤ε2‖∇z‖2 + Cε1,ε2A ‖z‖2

For simplicity set ε1 = ε2 (neither has been restricted yet), and plug the last twoinequalities into (108). After adjusting the constants we get

∣∣∣∣

∫

Ω

∫ 1

0

f ′′(λz + u)(λzt + ut) z2dλdΩ

∣∣∣∣≤ ε1CA ‖∇z‖2 + Cε1,A ‖z‖2. (109)

Now apply (109) and (103) to (102):∣∣∣∣

∫ t

s

(f(z), zt)

∣∣∣∣≤ ε[Ez(s) + Ez(t)] + CA ,ε,ε1

supθ∈[s,t]

‖z(θ)‖2 + (ε + ε1CA )

∫ t

s

‖∇z‖2

which is equivalent to the statement of Proposition 8, since parameters ε and ε1

estimate are arbitrary and independent of CA .

6.8.3. Proof of (47) and uniform smoothness of the attractor. For a pair of trajec-tories w and u through the attractor invoke Proposition 6 with initial time s ∈ R.Then use Proposition 8 to estimate the source terms (exactly as it was done in theproof of Proposition 7; relabel the constants accordingly):

Ez(s + T ) + CT ;1

∫ s+T

s

Ez(t)dt ≤σEz(s) + CA ,T,ε supθ∈[0,T ]

‖z(s + θ)‖2

+ εCT [Ez(s) + Ez(s + T )] + εCT

∫ s+T

s

Ez(t)dt.

(110)

Note that for sufficiently small ε the terms εCT (Ez(s)+Ez(s+T )) can be absorbedby Ez(s+T ) and σEz(t) (again, exactly as it was performed in the proof of Propo-

sition 7), whereas εCT

∫ s+T

sEz(t) can be absorbed into CT ;1

∫ s+T

sEz(t)dt. As a

result we have (possibly for a different σ = σ(ε) < 1)

Ez(s + T ) ≤ σEz(s) + CA ,T supθ∈[0,T ]

‖z(s + θ)‖2 ∀s ∈ R (111)

which completes the proof of (47) in Lemma 4.3.

To prove uniform bounds on the attractor in H2×H1 topology, use the statement(101) of Proposition 8 in the hypothesis (b) of Proposition 7. Then the resultingbound ‖utt(t)‖ + ‖ut(t)‖1,Ω ≤ CA (where CA depends only on the diameter of theattractor in the finite-energy topology) is independent of the trajectory u through

LONG-TERM DYNAMICS OF A WAVE EQUATION 497

the attractor, and holds for all t ∈ R. When combined with original equation (1)this estimate shows that ‖∆u‖ is uniformly bounded as well.

7. Appendix.

7.1. Uniform bounds on the energy and equilibria. We will prove that if theinitial data belongs to a bounded subset B of the finite energy space H , then thenorm of the evolution state is uniformly bounded in H by a constant, which onlydepends on the diameter of B. First recall that F ′ = f and

Ew(t) = Ew(t) −

∫

Ω

F (w(t))dΩ =1

2‖wt(t)‖

2 +1

2‖A1/2w(t)‖2 −

∫

Ω

F (w(t))dΩ

where A = −∆ is defined on H2(Ω) functions that satisfy appropriate boundaryconditions.

Also, let us repeat the energy identity originally stated in (38):

Ew(t) +

∫ t

s

∫

Ω

g(wt)wt = Ew(s). (a-112)

Proposition a-9 (Uniform energy bounds). Under Assumptions 1, 2 there existpositive constants c0, c1, dependent only on f and Ω, so that

c0Ew(t) − c1 ≤ Ew(t) ≤ Ew(0), ∀t ≥ 0.

Proof. By the virtue of energy identity (a-112) it suffices to only check the lowerbound on Ew(t). According to Assumption 1 (in particular (10)) there exist δ > 0and N = N(δ) > 0 such that f(s)/s ≤ (λ0 − δ) if |s| > N . Suppose s > N thenf(s) − (λ0 − δ)s ≤ 0; integrate:

0 ≥

∫ s

N

[f(t) − (λ0 − δ)t] dt = F (s) − F (N) −1

2(λ0 − δ)

[s2 − N2

]. (a-113)

Repeat this computation for s < −N (now f(s) ≥ −(λ0− δ)s and we integrate over]s,−N [ ); combine the result with (a-113):

F (s) ≤1

2(λ0 − δ) s2 + CF,N , |s| > N (a-114)

where CF,N = max |F (N)|, |F (−N)| + (λ0 − δ)N2/2. Therefore

1

2‖A1/2w‖2 −

∫

Ω

F (w) =1

2‖A1/2w‖2 −

∫

|w|≤N

F (w) −

∫

|w|>N

F (w)

≥ ε‖A1/2w‖2 + (1/2 − ε) ‖A1/2w‖2

−1

2(λ0 − δ)‖w‖2 − CF,N,Ω

with CF,N,Ω = m(Ω) · (CF,N + sup|s|≤N |F (s)|). By our choice of λ0 we have

‖A1/2w‖2 ≥ λ0 ‖w‖2, hence if 0 < ε < δ/(2λ0), we obtain

1

2‖A1/2w‖2 −

∫

Ω

F (w(t)) ≥ ε‖A1/2u‖2 − CF,N,Ω (a-115)

which implies the statement of the proposition for c0 = ε and c1 = CF,N,Ω.

The next proposition uses a very similar approach to show that the set N ofstationary points is bounded. Recall that every W = (w, 0) ∈ N satisfies equation

− ∆w = f(w) (a-116)

where w is subject to appropriate boundary conditions (B.C.-1,2, or 3).

498 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

Proposition a-10. When f(s) satisfies condition (10) (in Assumption 1), the setof solutions to (a-116) is bounded.

Proof. Recall that λ0 denotes the smallest eigenvalue of operator A. Multiply equa-tion (a-116) by w; then, according to (10), there exists some δ > 0 and a constantNδ > 0 such that

λ0

∫

Ω

w2(x)dx ≤ ‖A1/2w‖2 ≤ (λ0 − δ)

∫

S

w2(x)dx +

∫

Ω\S

f(w)wdx

where S = x ∈ Ω : |w(x)| ≥ Nδ. Since∣∣∫

Ω\Sf(w)w

∣∣ < CNδ,Ω, the integral over

S must be bounded independently of w.

7.2. Proof of the asymptotic smoothness: compensated compactness offunctional Ψ. The next proposition verifies the compensated compactness prop-erty of the lower-order and source-dependent terms in inequality (44). This resultproves condition (42) in the hypothesis of Proposition 1, which, in turn, when com-bined with (44) implies the asymptotic smoothness property of the flow (Theorem3.4).

Proposition a-11 (Compensated compactness of functional Ψ). Let z = w − u bethe difference of any two evolution trajectories with initial data Y1 = w0, w1, Y2 =u0, u1 in some bounded set B ⊂ H . Define

ΨB,T (Y1, Y2) := CT supt∈[0,T ]

‖z(t)‖2 + CT

∫ T

0

ds

∫ T

s

(f(z), zt) + CT

∫

QT

ξ2f(z)h · ∇z

where f(z) = f(w) − f(u). Assume f satisfies Assumption 1. In addition, adoptgeometrical conditions of Theorem 3.4, namely: Assumption 4 (when ΓU 6= ∅), andAssumption 5 (when ΓU 6= ∅ and either B.C.-2 with (16), or B.C.-3).

THEN, if the sequence Yn∞n=1 belongs to B, there exists a subsequence Ynk

such that

limmk→∞

limnk→∞

ΨB,T (Ymk, Ynk

) = 0. (a-117)

Proof. Consider a sequence of initial data Ynn∈N, Yn = w0n, w1n ⊂ B (Bbounded), with the corresponding evolution trajectories t 7→ wn(t), wn

t (t). Theidea of the proof is that some subsequence of Yn converges weakly in H , whileΨ exhibits certain compensated compactness due to its structure, and because thelimits in (a-117) are taken one at a time. Throughout the proof we will re-index allsubsequences by variables n and m without explicit mention.

¿From the uniform energy bound (Proposition a-9) conclude that on a subse-quence

wn → w weakly* in L∞(0, T ; H1(Ω))

wnt → wt weakly* in L∞(0, T ; L2(Ω)).

(a-118)

By (extended) Aubin’s compactness argument (e.g. see [44]) for the tower of spaces

L∞(0, T ; H1(Ω)) ⊂ L∞(0, T ; H1−η(Ω)) ⊂ L∞(0, T ; L2(Ω)), 0 < η ≤ 1

and due to (a-118), we know that (for a subsequence) wn, wnt → w, wt strongly

in L∞(0, T ; H1−η(Ω)). However, since each t 7→ wn(t), wnt (t) is continuous into

H , we further have

wn → w in C([0, T ]; H1−η(Ω)). (a-119)

LONG-TERM DYNAMICS OF A WAVE EQUATION 499

Introduce zn,m = wn − wm. Now we will verify the limit condition for each of thesummands in the definition of Ψ.

(i) From the convergence (a-119) we immediately have

supt∈[0,T ]

‖zn,m‖1−η,Ω → 0 as n, m → ∞ (a-120)

(ii) Next, note the following decomposition:∫ T

0

ds

∫ T

s

(f(zn,m), zn,m

t

)=T

∫

Ω

[F (wn(T )) + F (wm(T ))

]dΩ

−

∫

QT

[F (wn) + F (wm)

]

−

∫ T

0

ds

∫ T

s

[(f(wn), wm

t ) + (f(wm), wnt )]

(a-121)

We may pass to a subsequence such that:• wn(T ) → w(T ) a.e. in Ω• wn → w a.e. in QT

• and (by (13)) f(wn) converges weakly* to some element in L∞(0, T ; L2(Ω))By Egoroff’s theorem we can find a set Ωδ ⊂ Ω of arbitrarily small measure

δ > 0 such that F (wn(T )) converges to F (w(T )) uniformly on Ω \Ωδ. On Ωδ

use the fact that F (s) = O(s4) is subcritical for the Sobolev embedding H1 →L1, in particular, F (w) is bounded in L∞(0, T ; L3/2(Ω)), and by Holder’sinequality

∫

Ωδ

|F (wn(T )) − F (w)| ≤δ ‖F (wn(T )) − F (w(T ))‖L3/2(Ω)

≤δ CB → 0 as δ ց 0.

Combine the convergence on Ωδ and Ω \ Ωδ to find

F (wn(T )) → F (w(T )) in L1(Ω). (a-122)

Now apply this procedure to F (wn) on the set QT . We have pointwise a.e.convergence F (wn) → F (w) on QT , whereas on a subset of small measure δuse

∫

QT δ

|F (wn) − F (w)| ≤δ ‖F (wn) − F (w)‖L3/2(QT δ)

≤ δ T 2/3‖F‖L∞(0,T ;L3/2(Ω)) ≤ δCB,T → 0 as δ ց 0

whence

F (wn) → F (w) in L1(QT ). (a-123)

Then use exactly the same argument again to show that f(wn) → f(w) inL1(QT ). Since f(wn) also converges weakly in L2(QT ), then the weak limitmust coincide with f(w):

f(wn) → f(w) weakly* in L∞(0, T ; L2(Ω)). (a-124)

Therefore as n → ∞,∫ T

0

ds

∫ T

s

∫

Ω

| (f(wn) − f(w), wmt ) | ≤ T

∫

QT

|f(wn) − f(w)| |wmt | → 0. (a-125)

500 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

Now invoke (a-125) and the weak convergence wmt → wt in L2(QT ) to conclude

limn→∞

limm→∞

∫ T

0

ds

∫ T

s

[(f(wn), wmt ) + (f(wm), wn

t )] =

= 2

∫ T

0

ds

∫

Ω

[F (w(T )) − F (w(s))].

(a-126)

Results (a-122) and (a-126) verify that the LHS of (a-121) converges to 0 onsome subsequence of wn.

(iii) An approach similar to the one used in item (ii) above applies to the lastintegral in the definition of Ψ.

∫

QT

ξ2h · f(zn,m)∇zn,m =

∫

QT

ξ2h ·[∇F (wn) + ∇F (wm)

]dΩ

−

∫ T

0

(ξ2f(wn), h · ∇wm

)−

∫ T

0

(ξ2f(wm), h · ∇wn

)

=

∫

ΣT

ξ2(h · ν)[F (wn) + F (wm)]

−

∫

QT

ξ2(div h)[F (wn) + F (wm)

]dΩ

−

∫ T

0

(ξ2f(wn), h · ∇wm

)−

∫ T

0

(ξ2f(wm), h · ∇wn

)

(a-127)

Note that ∇F (wn) = f(wn)∇wn ∈ L1(Ω) whence we may use Sobolev em-bedding W 1,1(Ω) → L1(Γ) (e.g. see [1, Thm. 5.22]) to justify the abovedecomposition.

Recall that ξ is only supported on ΓU . So if the homogeneous Dirichletcondition holds on ΓU , the boundary terms on the RHS of (a-127) vanish.Likewise, if ΓU is subject to Neumann-type dynamics, then condition h ·ν = 0,which follows from Assumption 5 (e.g. see (28)) applies, and the traces in(a-127) disappear as well.

Next, because ξ and |h| are uniformly bounded, we can use (a-123) toconclude

limm→∞

limn→∞

∫

QT

ξ2(div h)[F (wn) + F (wm)

]= 2

∫

QT

ξ2(div h)F (w).

Similarly, (a-118) and (a-124) imply

limm→∞

limn→∞

∫ T

0

(ξ2f(wn), h · ∇wm

)=

∫ T

0

(ξ2f(w), h · ∇w

)

=

∫

QT

ξ2h · ∇F (w) = −

∫

QT

ξ2(div h)F (w).

Combination of the last two limits gives

limm→∞

limn→∞

∫

QT

ξ2h · f(zn,m)∇zn,m = 0.

Thus, all the terms in the definition of Ψ satisfy the desired condition (a-117).

LONG-TERM DYNAMICS OF A WAVE EQUATION 501

7.3. Carleman multipliers (proof of Proposition 5). Below we derive funda-mental identity (64) stated in Proposition 5. The calculations are self-contained andhold for any equation of the form (1), provided solutions w, wt have regularity atleast L∞(0, T ; H3/2+ε(Ω) × H1(Ω)).

Identity (64) does not make use of the structure of the cutoff maps ζ, ξ (Section1.4.1) or properties of the vector field h (Section 1.4.2). So for the purposes of thesubsequent argument we may declare these objects to be fairly arbitrary:

• Let c, T be any positive constants and d(x) be any smooth function on Ω.Define

Φ(t, x) = d(x) − c

∣∣∣∣t −

T

2

∣∣∣∣

2

, h := ∇d.

Thus ∇Φ is time-independent. Also note that Φt is space-independent, with

Φt = −2c

(

t −T

2

)

= c (T − 2t) , Φtt = −2c, Φt(T ) = −cT Φt(0) = cT.

(a-128)We define Jh to be the Jacobian of h (i.e. the Hessian of d(x)). The com-putations will not make use of the fact that Jh is strictly positive definite(e.g. Jh ≥ ρI as in Section 1.4.2), so, for now, constant ρ can be assigned anarbitrary positive value.

• Suppose w, wt, u, ut are strong solutions, with initial data respectivelyw0, w1 and u0, u1. Set z = w − u.

• Assume that ζ, ξ ∈ C2(Ω); for this section, no additional assertions need tobe imposed on these maps.

• Let z = ξz and z = ζz. Multiplication of the difference equation (8) by ξgives

ztt − ∆z + [[∆, Mξ]]z + ξχg(zt) = ξf(z) in QT (a-129)

where Mξ denotes pointwise (a.e.) multiplication by ξ. Similarly for z wehave

ztt − ∆z + [[∆, Mζ ]]z + ζχg(zt) = ζf(z) in QT . (a-130)

7.3.1. Multiplier M1. As in [36], we construct the first multiplier:

M1 = M1(Φ, τ ; z, zt) := eτΦ[h · ∇z − Φtzt]. (a-131)

Use M1 as the test function in the strong formulation of (a-129), and integrate therelation in time. We will expand the result term by term.

Product with ztt

∫

QT

ztteτΦ[h · ∇z − Φtzt] =

∫

QT

eτΦztth · ∇z −1

2

∫

QT

eτΦΦtd

dt

(z2

t

)

=

[∫

Ω

eτΦzth · ∇z

]T

0

−

∫

QT

ztd

dt

(eτΦh · ∇z

)

−1

2

[∫

Ω

eτΦΦtz2t

]T

0

+1

2

∫

QT

z2t

d

dt

(eτΦΦt

).

(a-132)

502 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

Use (a-128) to expand the terminal quantities in (a-132):

k1;0,T (z, zt) :=

[∫

Ω

eτΦzt

(

h · ∇z −1

2Φtzt

)]T

0

=

∫

Ω

eτΦ(x,T )zt(T )

(

h · ∇z(T ) +cT

2zt(T )

)

−

∫

Ω

eτΦ(x,0)zt(0)

(

h · ∇z(0) +cT

2zt(0)

)

.

(a-133)

In addition∫

QT

ztd

dt

(eτΦh · ∇z

)=

∫

QT

zth · (τeτΦΦt∇z + eτΦ∇zt)

=τ

∫

QT

eτΦztΦth · ∇z +1

2

∫

ΣT

eτΦz2t (h · ν)

−1

2

∫

QT

z2t div (eτΦh).

(a-134)

Substitute the last relation into (a-132):∫

QT

zttM1 =1

2

∫

QT

z2t

[d

dt(eτΦΦt) + div (eτΦh)

]

− τ

∫

QT

eτΦztΦth · ∇z

−1

2

∫

ΣT

eτΦz2t (h · ν) + k1;0,T .

(a-135)

Product with −∆zThe term ∆z yields∫

QT

(−∆z)eτΦ[h · ∇z − Φtzt] = −

∫

ΣT

∂z

∂νM1 + τ

∫

QT

(∇z · h)M1

+

∫

QT

eτΦ∇z · ∇(h · ∇z) −

∫

QT

ΦteτΦ∇z · ∇zt.

(a-136)

Now recall that Jh denotes the Jacobian of h, and invoke the identity

∇v · ∇(h · ∇v) = (Jh∇v) · ∇v +1

2h · ∇|∇v|2 (a-137)

to obtain∫

QT

eτΦ∇z · ∇(h · ∇z) =

∫

QT

eτΦ(Jh∇z) · ∇z +1

2

∫

QT

eτΦh · ∇|∇z|2

=

∫

QT

eτΦ(Jh∇z) · ∇z +1

2

∫

ΣT

eτΦ|∇z|2(h · ν)

−1

2

∫

QT

div (eτΦh)|∇z|2.

(a-138)

Next,∫

QT

ΦteτΦ∇z · ∇zt =

1

2

∫

QT

ΦteτΦ d

dt|∇z|2 =

1

2

[∫

Ω

ΦteτΦ|∇z|2

]T

0

−1

2

∫

QT

|∇z|2d

dt

(Φte

τΦ).

(a-139)

LONG-TERM DYNAMICS OF A WAVE EQUATION 503

Introduce another shorthand for the terminal products

k2;0,T = k2;0,T (z, zt) :=1

2

[∫

Ω

ΦteτΦ|∇z|2

]T

0

= −cT

2

∫

Ω

eτΦ(x,T )|∇z(T )|2

−cT

2

∫

Ω

eτΦ(x,0)|∇z(0)|2.

(a-140)

Equations (a-138), (a-139), (a-140) permit to rewrite (a-136) as

∫

QT

(−∆z)M1 = −

∫

ΣT

∂z

∂νM1 +

1

2

∫

ΣT

eτΦ|∇z|2(h · ν)

+

∫

QT

eτΦ(Jh∇z) · ∇z −1

2

∫

QT

|∇z|2[

div (eτΦh) −d

dt

(Φte

τΦ)]

+ τ

∫

QT

(h · ∇z)M1 − k2;0,T .

(a-141)

Result of the multiplication by M1

Put together (a-135) and (a-141) to establish that when the strong formulation ofthe (a-129) acts on the test function M1 = eτΦ[∇Φ ·∇z−Φtzt] = eτΦ[h ·∇z−Φtzt]it produces the identity:

∫

ΣT

∂z

∂νM1 +

1

2

∫

ΣT

eτΦ(z2

t − |∇z|2)(h · ν) =

1

2

∫

QT

(z2

t + |∇z|2) d

dt(eτΦΦt) +

1

2

∫

QT

(z2

t − |∇z|2)div (eτΦ∇Φ)

+

∫

QT

eτΦ(Jh∇z) · ∇z

+ τ

∫

QT

(h · ∇z)(M1 − eτΦztΦt)

+

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z))

M1

+ k1;0,T − k2;0,T

(a-142)

Remark 19. Let us briefly elaborate on the role of the Carleman weight eτΦ

in the structure of equation (a-142). The map eτΦ takes advantage of the higherregularity of z, in particular of the presence (in the original equation) of second orderderivatives ztt and ∆z. Since finite energy only involves first-order differentials we“discard” one derivative in space and one in time via Green’s identities. However,the exponential eτΦ “counts” the number of extra derivatives by leaving (energy-level) commutators for each application of Green’s formula. I.e. commuting eτΦ

past the differential operators produces additional terms with a factor of τ in front(originated from differentiation of eτΦ). On the other side of the equation, no factor

of τ will show up since perturbation f(z) already resides at the ground energy level.

By increasing τ we will eventually be able to absorb the source eτΦf(z) into theenergy terms.

504 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

7.3.2. Equipartition of energy: multiplier M2 = z m(t, x). To motivate the subse-quent derivations we outline some immediate goals:

1. The next multiplier exhibits a relation between the kinetic ‖zt‖2 and potentialcomponents ‖∇z‖2 of the quadratic energy functional. This equation is oftencalled the “equipartition” of energy, as in the absence of a destabilizing per-turbations the kinetic and potential constituents are topologically equivalentwith respect to the ground level H ⊂ H1(Ω) × L2(Ω).

2. Use the equipartition relation to “disentangle” energy level terms in (a-142)from the factor div (eτΦ∇Φ).

3. Then invoke the equipartition again to recover full quadratic energy from thepotential component

∫

QTeτΦ(Jh∇z) · ∇z on the RHS of (a-142).

Now let m(x, t) be some C1 function defined on QT . Later we will specify severaluseful candidates for m(x, t). Introduce the second multiplier

M2(m) = M2(z, m(x, t)) := zm. (a-143)

Multiply equation (a-129) by M2(m) and integrate over the space-time cylinderQT . Some of the terms in the resulting inequality can be expanded as follows:

∫

QT

zttzm =

[∫

Ω

ztzm

]T

0

−

∫

QT

(z2t m + ztzmt)

∫

QT

(∆z) zm =

∫

ΣT

∂z

∂νzm −

∫

QT

(|∇z|2m + z∇z · ∇m).

Hence the second multiplier yields (after some rearrangement of the terms)∫

QT

[z2t − |∇z|2]m = −

∫

ΣT

(∂z

∂ν

)

zm +

∫

QT

z(∇z · ∇m − ztmt)

+

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z))

zm +

[∫

Ω

ztzm

]T

0

(a-144)

7.3.3. Isolating the energy level terms. Define

µ(x, t) := div (eτΦh) −d

dt

(eτΦΦt

)= eτΦ

(τ |h|2 − τΦ2

t + div h + 2c). (a-145)

Substitute m = µ(x, t) into the identity (a-144), then multiply the expression by 12

and rearrange:

1

2

∫

QT

(z2

t − |∇z|2)div (eτΦh) =

1

2

∫

QT

(z2

t − |∇z|2) d

dt

(eτΦΦt

)−

1

2

∫

ΣT

∂z

∂νzµ

+1

2

∫

QT

z (∇z · ∇µ − ztµt)

+1

2

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z))

zµ

+1

2

[∫

Ω

ztzµ

]T

0

.

(a-146)

Let

k3;0,T := k1;0,T − k2;0,T +1

2

[∫

Ω

ztzµ

]T

0

. (a-147)

LONG-TERM DYNAMICS OF A WAVE EQUATION 505

Recalling the definition (a-131) of M1 we have

∫

QT

z2t

eτΦ(τΦ2t−2c)

︷ ︸︸ ︷

d

dt(eτΦΦt) + τ

∫

QT

(h · ∇z)(M1 − eτΦztΦt)

=τ

∫

QT

eτΦz2t Φ2

t + τ

∫

QT

e−τΦ(M1 + eτΦztΦt)(M1 − eτΦztΦt)

− 2c

∫

QT

eτΦz2t

=τ

∫

QT

e−τΦM21 − 2c

∫

QT

eτΦz2t .

(a-148)

Substitute (a-146) into (a-142), then apply (a-147) and (a-148) to the result. Inaddition add and subtract the term

∫

QTρeτΦ|∇z2| on the RHS of the resulting

equation; as far as the fundamental identity is concerned the value of ρ is irrelevant;however, eventually we will use the one stated in relation (29). We finally come to∫

ΣT

∂z

∂νM1 +

1

2

∫

ΣT

∂z

∂νzµ +

1

2

∫

ΣT

eτΦ(z2

t − |∇z|2)(h · ν) =

=

∫

QT

eτΦ(Jh − ρId)∇z · ∇z

+

∫

QT

ρeτΦ|∇z|2 − 2c

∫

QT

eτΦz2t

+1

2

∫

QT

z(∇z · ∇µ − ztµt) + τ

∫

QT

e−τΦM21

+

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z))( zµ

2+ M1

)

+ k3;0,T

(a-149)

7.3.4. Reconstructing the quadratic energy from the potential component on supp ξ.The next step recovers both components, ‖∇z‖2 and ‖zt‖2, of the quadratic energy(however only on supp ξ) from the integral

∫

QTeτΦ(Jh∇z) · ∇z in (a-149). Begin

with (a-149) and split the newly added term∫

QTρeτΦ|∇z2| into two summands:

ρ

∫

QT

eτΦ|∇z|2 − 2c

∫

QT

eτΦz2t =(ρ/2 − c)

∫

QT

eτΦ(|∇z|2 + z2t )

+ (ρ/2 + c)

∫

QT

eτΦ(|∇z|2 − z2t ).

(a-150)

Call upon the equipartition identity (a-144) with m = −eτΦ:∫

QT

(|∇z|2 − z2t )eτΦ =

∫

ΣT

(∂z

∂ν

)

zeτΦ −

∫

QT

z

(

∇z · ∇eτΦ + ztd

dt(eτΦ)

)

−

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z))

zeτΦ −

[∫

Ω

ztzeτΦ

]T

0

.

(a-151)

506 I. CHUESHOV, I. LASIECKA AND D. TOUNDYKOV

Substitute (a-151) in place of the corresponding integral on the RHS of (a-150).Then use the outcome to replace the difference ρ

∫

QTeτΦ|∇z|2 − 2c

∫

QTeτΦz2

t in

(a-149).Before stating the result group together some of the terms:

(BΣ)τ :=

∫

ΣT

∂z

∂νM1+

∫

ΣT

∂z

∂νz

[1

2µ − (ρ/2 + c)eτΦ

]

+1

2

∫

ΣT

eτΦ(z2

t − |∇z|2)(h·ν)

(a-152)

k0,T := k3;0,T − (ρ/2 + c)

[∫

Ω

ztzeτΦ

]T

0

. (a-153)

We have the first half of the fundamental identity:

(BΣ)τ =

∫

QT

eτΦ(Jh − ρId)∇z · ∇z + (ρ/2 − c)

∫

QT

eτΦ(|∇z|2 + z2t )

+

∫

QT

z∇z · ∇(µ

2− (ρ/2 + c)eτΦ

)

−

∫

QT

z ztd

dt

(µ

2− (ρ/2 + c)eτΦ

)

+ τ

∫

QT

e−τΦM21

+

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z)) [

M1 + z(µ

2− (ρ/2 + c)eτΦ

)]

+ k0,T

(a-154)

7.3.5. Energy over supp ζ. So far we have collected the energy over supp ξ. Now wewill derive the corresponding equation in variable z. The version of equipartitionrelation for the equation in z (a-130), is analogous to the one for z (a-144): simplyreplace z with z and ξ with ζ; use this identity with m = −eτΦ:

∫

QT

(|∇z|2 − z2t )eτΦ =

∫

ΣT

(∂z

∂ν

)

zeτΦ −

∫

QT

z

(

∇z · ∇eτΦ − ztd

dt(eτΦ)

)

−

∫

QT

(

ζχg(zt) + [[∆, Mζ]]z − ζf(z))

zeτΦ −

[∫

Ω

ztzeτΦ

]T

0

.

(a-155)

As before, create a short-hand for the terminal products:

k0,T :=

[∫

Ω

ztzeτΦ

]T

0

. (a-156)

Add the kinetic term 2∫

QTeτΦz2

t on each side of (a-155) and multiply the entire

identity by some constant C (for these calculations C can be arbitrary, in the proofs

LONG-TERM DYNAMICS OF A WAVE EQUATION 507

of the main results we will eventually assign C a suitably large value).

C

∫

QT

(|∇z|2 + z2t )eτΦ =C

∫

ΣT

(∂z

∂ν

)

zeτΦ + 2C

∫

QT

eτΦz2t

− C

∫

QT

z

(

∇z · ∇eτΦ − ztd

dteτΦ

)

− C

∫

QT

(

ζχg(zt) + [[∆, Mζ ]]z − ζf(z))

eτΦz − Ck0,T

(a-157)

Add this equation to (a-154) to acquire the fundamental identity that connectsboundary and interior dynamics. After we group the spatial traces via

(BΣ)τ := (BΣ)τ + C

∫

ΣT

(∂z

∂ν

)

zeτΦ

the fundamental identity reads

(BΣ)τ =

∫

QT

eτΦ(Jh − ρId)∇z · ∇z + (ρ/2 − c)

∫

QT

eτΦ(|∇z|2 + z2t )

+ C

∫

QT

eτΦ(|∇z|2 + z2t ) − 2C

∫

QT

eτΦz2t

+

∫

QT

z∇z · ∇(µ

2− (ρ/2 + c)eτΦ

)

+ C

∫

QT

z∇z · (∇eτΦ)

−

∫

QT

z ztd

dt

(µ

2− (ρ/2 + c)eτΦ

)

− C

∫

QT

zztd

dt(eτΦ)

+ τ

∫

QT

e−τΦM21

+

∫

QT

(

ξχg(zt) + [[∆, Mξ]]z − ξf(z)) [

M1 + z(µ

2− (ρ/2 + c)eτΦ

)]

+ C

∫

QT

(

ζχg(zt) + [[∆, Mζ ]]z − ζf(z))

eτΦz

+ k0,T + Ck0,T .

(a-158)

A mere rearrangement of terms in (a-158) yields the statement (64) of Proposition5.

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Received January 2007; revised August 2007.

E-mail address: il2v@virginia.edu

E-mail address: chueshov@univer.kharkov.ua

E-mail address: dtoundykov2@math.unl.edu