Finite dimensional attractor for a composite system of wave/plate equations with localised damping

0.49 MB·English

by  Francesca Bucci

FINITE DIMENSIONAL ATTRACTOR FOR A COMPOSITE
SYSTEM OF WAVE/PLATE EQUATIONS WITH LOCALISED
0 DAMPING
1
0
2 Francesca Bucci

n
Universit`adegliStudidiFirenze
a
DipartimentodiMatematicaApplicata
J
Firenze,50139, Italy
7
2 Daniel Toundykov

] UniversityofNebraska–Lincoln
S DepartmentofMathematics
D Lincoln,NE68588, USA
h. Abstract. The long-term behaviour of solutions to a model for acoustic-
t structureinteractionsisaddressed;thesystemiscomprisedofcoupledsemilin-
a earwave(3D)andplateequationswithnonlineardampingandcriticalsources.
m
The questions of interest are: existence of a global attractor for the dynamics
[ generatedbythiscompositesystem,aswellasdimensionalityandregularityof
theattractor. Adistinctandchallengingfeatureoftheproblemisthegeomet-

2 ricallyrestricteddissipationonthewavecomponentofthesystem. Itisshown
v thattheexistence ofaglobalattractor offinitefractaldimension—established
4 inapreviousworkbyBucci,ChueshovandLasiecka(Comm. PureAppl. Anal.,
6 2007)onlyinthepresenceoffullinterior acousticdamping—holdseveninthe
4 case of localised dissipation. This nontrivial generalization is inspired by and
5 consistent with the recent advances in the study of wave equations with non-
. linearlocaliseddamping.
2
1 1. Introduction
9
0
This paper focuses on the long-termbehaviour of a system of partialdifferential
:
v equations (PDE’s, or simply PDE) modeling acoustic waves in a three-dimensional
i domain Ω, and their interaction with the elastic vibrations of a part Γ Γ:=∂Ω
X 0 ⊂
of the boundary. The corresponding PDE problem ((2.1) below) is comprised of
r
a semilinear wave and plate equations on domains Ω and Γ0 respectively; a detailed
description of the system will be given in the next section.
PDE models for acoustic-structure interactions have received a great deal of at-
tention in the past decade, mainly in the context of control theory, in connection
with(butnotlimitedto)theproblemofreducingthenoiseinsideanacousticcham-
ber and stabilizing its vibrating walls. As in the case of a single PDE, the sought-
afterproperties—onatheoreticallevel,yetwithimplicationsforthecomputational
This research was partly performed while both authors were visiting the Centro di Ricerca
MatematicaEnnioDeGiorgioftheScuolaNormaleSuperioreinPisa,withintheprogramResearch
inpairs. Thesupport(andkindness)oftheDirectorMarianoGiaquinta,aswellastheco-operative
assistanceofthesecretarialstaffoftheCentroarewarmlyacknowledged.
TheresearchofF.BucciwaspartiallysupportedbytheUniversit`adegliStudidiFirenze,and
bytheItalianMIUR,withintheproject2007WECYEA (“Metodi diviscosita`,metriciediteoria
delcontrolloinequazioniallederivateparzialinonlineari”).
The research of D. Toundykov was partially supported by the National Science Foundation
underGrantDMS-0908270.
1
2 F.BUCCI,D.TOUNDYKOV
methods—are (besides well-posedness): stability, controllability, and solvability of
the associated optimal control problems. However, the presence of two (or more)
coupledequations,of different type (usually, hyperbolic andparabolic)and/oract-
ing onmanifolds of differentdimensions,rendersthe PDE analysisof the dynamics
far more complex.
It is beyond this work’s scope to provide a comprehensive account of the literature
on control problems for such interconnected PDE systems. However, for complete-
ness and the reader’s convenience some notable contributions are listed below.
Deep insight into the physical origin of PDE models for acoustic-structure in-
teractions is provided by [47]. A more recent reference on analytical methods for
modeling acoustic problems is [36].
A general reference for the mathematical control theory of established coupled
PDE systems for acoustic-structureinteractionsis the treatise[40], whichalsocon-
tainsanextensivebibliography,fromwhichweexplicitlycite[2](PDEanalysisand
optimalcontrol),[3,4](controllability),and[45,41](uniformstability),besidesthe
former [25], [8], [46].
More recent advances include, for instance (without any claim of completeness),
[42], [44] and [1] (general theory of quadratic optimal control and of Riccati equa-
tions), [9, 10] (sharpinterior/boundaryregularity,with applicationto optimalcon-
trol), [12] (stabilization). For a survey of results on exact boundary controllability
and uniform boundary stabilizability by differential geometric methods, see [34].
Thestabilityofaninteresting2Dstructural-acousticmodelwasdiscussedin[32,33].
The focus of this study is on the long-term behavior of the nonlinear dynamics
generated by the system (2.1) below, such as existence of a global attractor and
its properties: geometry, fractal dimension, regularity. It was shown in [11] that
in the presence of an acoustic dissipation distributed over the entire domain Ω,
existence of a global attractor is guaranteed under some basic assumptions on all
the nonlinearities,including the mostrelevantcasewhenthe functionmodelingthe
acousticsourcehasacritical exponent. Furthermore,evenunderacritical,i.e.non-
compact, perturbation the attractor has a finite fractal dimension, provided the
dissipationtermsactingoneitherequationsatisfysuitableconditions;inparticular,
the acoustic dissipation must be linearly bounded.
In light of the recent advances in the study of the long-term behaviour of wave
equations with nonlinear damping [21, 22], the present work aims at showing that
dissipationactiveonlyinaneighbourhoodofΓ —combinedwithanonlinearinterior
0
platedamping,asinthePDEmodeladdressedin[11]—stillguaranteestheexistence
of a finite-dimensional global attractor.
With this as background motivation, let us now discuss the major challenges
which arise in the present context of
coupled hyperbolic PDE’s (wave and plate equations),
•
localiseddampinginconjunctionwithacriticalexponent-source onthewave
•
equation,
a critical-exponent source on the plate equation,
•
andthendescribetheprincipaltheoreticalandtechnicalargumentsutilizedtoover-
come them.
1.1. New challenges. Critical sources alone have been well-known to present an
intrinsic obstacle to formation of attractors. At the level of critical-exponents the
compactness of corresponding Sobolev embeddings is lost, and “critical terms” are
ATTRACTORS FOR A COUPLED WAVE/PLATE SYSTEM 3
essentially non-compact perturbations of the principal dynamics. Thus, in general,
one would not expect the flow to converge to a compact set, especially in hyper-
bolic systems. For an overviewof associatedchallenges see the very comprehensive
treatise [20]. The recent advances in this direction, however, predominantly rely
onthe dissipationmechanismbeing supportedon the entire domain. In particular,
for structural acoustic interactions, it was shown in [11] that in the presence of
an acoustic dissipation distributed over the whole domain domain Ω, is sufficient
for existence of a global finite-dimensional attractor, even in presence of a critical
acoustic source.
However, the physically relevant localised interior (or boundary) damping poses
yet further difficulties, since besides the lack of compactness from criticality, the
dissipationnow mustbe “propagated”acrossthe domaininorderto haveanykind
of a stabilizing effect. A critical source and geometrically restricted damping in
a single wave equation had been a long-standing open problem, whose solution
ultimately necessitated Carleman estimates and geometric optics analysis [21, 22].
Ina composite setting the aforementionedtechniques wouldhavetoaccountnot
merelyfortwodifferenttypesofdynamics,butalsofortheNeumann-typecoupling
on the interface. Moreover, the source on the plate is described by higher-order
operators and, if approached via the same strategy as the acoustic counterpart,
presents a much more formidable obstacle. To overcome this difficulty, we develop
a new “hybrid” method that employs different techniques applicable to waveswith
localised damping, while taking advantage of the full interior dissipation on the
plate;atthe sametimetheanalysisaccommodatesthe mixedtermsthatarisefrom
the coupling.
Ultimatelyitfollowsthatacousticdampingactiveonly in aneighbourhood ofthe
flexible boundary—combined with a nonlinear interior plate damping—still guar-
antees the existence of a finite-dimensional global attractor, under “minimal” as-
sumptions on the nonlinear functions; “minimal” here indicating those consistent
with the hypotheses which yield a global attractor for the dynamics generated by
a single wave equation.
1.2. Outline of the argument. Toachieveourgoalwewillappealtoseveralgen-
eralcriteriafromthe theoryofinfinite-dimensionaldynamicalsystems. While well-
posedness follows via the classical theory of monotone operators in [6], along with
somerecentadditionsdevisedin[16],theasymptoticanalysiswillbenefitfromnovel
abstractresults gatheredin[20], specifically tailoredfor dissipative evolutionequa-
tions of hyperbolic type. (For general references on dissipative infinite-dimensional
systems, the reader is referred to the cornerstones: [5], [53] and [35]; see also [39]
and[14],thelatteraddressingnon-autonomoussystems.) Morespecifically,weshall
invoke
(i) a compactness criterion—introduced in [37] and subsequently recast into a
moregeneralabstractformin[19]—recordedhereasTheoremA.2(thisresult
is central to the proof of existence of a global attractor),
(ii) a generalization of Ladyzhenskaia’s Theorem on dimension of invariant sets,
thatisTheoremA.4, whichwillfinallyenable us toinfer finite-dimensionality
of the attractor.
The application of the aforesaid criteria is not immediate, as one might guess
in light of the analysis already carried out on the composite PDE system in [11]
with (acoustic) full interior damping, and of the novel tools devised in [21] for the
4 F.BUCCI,D.TOUNDYKOV
(uncoupled)waveequationwithlocalised dissipation. Aprimarysourceofdifficulty
stems from the fact that these abstract results require estimates on the quadratic
energy corresponding to the difference of two trajectories, rather than to a single
trajectory,andthepresenceoflocalisedacousticdissipationinconjunctionwiththe
coupling (with the plate equation) brings about further technical difficulties, over
those encountered in [11] and [21], as explained in 1.1.
§
The analysis to be carried out includes various steps. After a preliminary dis-
cussionofwell-posednessofthe PDEsysteminanappropriatestate space,amajor
task will be to establish asymptotic compactness. That the global attractor has
a finite fractal dimension will be obtained subsequently, through a use of both a
Carleman-like identity established in [21] for the wave dynamics, and exploiting
the attractor’s compactness. As expected, proving finite dimensionality and the
additional smoothness of the attractor will require stronger assumptions on the
nonlinear dissipation terms in either of the coupled equations.
Note that we chose to separate the proof of existence of a global attractor from
that of its finite-dimensionality, as the former property does hold under weaker as-
sumptions. Inthis respect,the presentargumentdiffers fromthe onesfollowedina
largepartoftherecentliteratureondissipativedynamicalsystems,whereexistence
of an exponential1 attractoris sought, as it brings about—besides a certain robust-
ness under perturbations and numerical approximations—the intrinsic feature of
finite dimensionality; see, e.g. (and without any claim of completeness), [27], [28],
[29], [30], [31], [50]. The question whether the PDE system under investigation
possesses an exponential attractor will not be discussed here.
We finally note that the boundedness of the attractor in a smoother functional
space is closely tied to its finite-dimensionality. The proofs of both results are
interconnected; it is precisely the compactness of the attractor in the finite-energy
space and the regularity estimate in the higher-energy space which together show
that critical (non-compact) sources, do not substantially perturb the structure of
the attracting set. A recent result [24] provides an elegant way to prove regularity
ofaglobalattractorinhighernorms,withoutdirectlyappealingtouniform-in-time
estimates; however, the application of this argument to a wave equation requires a
stronger damping than employed in the present case.
A brief outline of the paper follows.
In Section 2 we introduce the PDE model under investigation, along with
•
thestatementsofourmainresults: well-posedness(Theorem2.4),existence
of a global attractor for the corresponding dynamics (Theorem 2.5), and
finite-dimensionality, as well as regularity of the attractor (Theorem 2.7).
The natural energies and state spaces for either uncoupled equation, and
then for the coupled PDE system, are introduced and briefly discussed in
Section 2.2.
Section 3 contains some preliminary work for the proof of existence of the
•
attractor. Moreprecisely,weestablishabasicidentityinvolvingtheintegral
of the quadratic wave and plate energies pertaining to the differences of
trajectories; see Lemma 3.4.
Section 4 is devoted to the proof of Theorem 2.5. Existence of a global
•
attractor is established in view of the gradient structure of the dynamical
system and asymptotic smoothness of the semi-flow. In turn, this latter
1Theconcept ofexponential attractor wasintroducedin[26].
ATTRACTORS FOR A COUPLED WAVE/PLATE SYSTEM 5
(crucial) property follows from a pointwise estimate of the total quadratic
energy (Proposition 4.2), combined with a suitable “weak sequential com-
pactness” property (Proposition 4.3).
Section 5 is focused on the proof of Theorem 2.7, technically based on the
•
keyinequalityestablishedinLemma5.1(refinedinLemma5.3),andonthe
Proposition 5.4. Finally, a short Appendix concludes the paper.
2. The PDE model and the statement of main results
The PDE system under investigation models structure-acoustics interactions
described by the acoustic velocity potential ζ(x,t) evolving in a smooth domain
Ω R3, and the vertical displacement w(x,t) of a thin hinged plate whose mid-
⊂
section occupies a part of the boundary of Ω; we denote the latter by Γ := ∂Ω,
and assert that it consists of two relatively open simply-connected segments with
overlapping closures Γ Γ , where Γ represents the plate, and Γ is the rigid
0 1 0 1
∪
wall of the acoustic chamber. The acoustic damping is confined to a subset of Ω
corresponding to the support of the cutoff function χ(x).
ζ ∆ζ+χ(x)g(ζ )=f(ζ) in Ω (0,T)
tt t
− ×
∂ζ
 =0 on Γ (0,T)
(2.1)  ζ∂∂∂(ννζ0,=·)α=κζw0t, ζt(0,·)=ζ1 oinn ΩΓ10××(0,T)
w +∆2w+b(w )+βκζ =f (w) in Γ (0,T)
tt t t|Γ0 2 0×
The positive cwwon(=·s,t0a∆)nw=ts=wα0,0,β awntd(·,κ0)re=prwes1ent parametionenrs dΓ∂iΓ0ct0.a×ted(0,bTy)the physical
model in question. Following [11], we shall consider, specifically,
(2.2) f (w)=p Q w2 ∆w,
2 0
− − |∇ |
h ZΓ0 i
which is the semilinear term that occurs in Berger’s plate model: the function p ,
0
which is related to transversalforces, belongs to L (Γ ), while the real constant Q
2 0
describes in-plane forces applied to the plate; for more on modeling according to
the Berger approach, see [15].
Remark1. Itisimportanttoemphasisethatalthoughweconsideraplateequation
with a nonlinear term f of the form (2.2), the analysis carried out in the present
2
papermightbe extendedtomoregeneralnonlinearfunctions, providedtheysatisfy
properconditions,suchasthoselistedinAssumption4.11in[20](followingthisref-
erence’s notation,in the presentcase woulddenote the plate dynamics operator,
A
that is the realization of the bilaplacian ∆2 with hinged boundary conditions).
2.1. Assumptionsandmainresults. Duetogeometricrestrictionsontheacous-
ticdamping,theshapeofΩmustsatisfycertainrestrictionsinorderforthefeedback
χ(x)g(ζ ) to be “effective,” as dictated by geometric optics andthe classicalresults
t
on propagationof singularities [7]. A sufficient assumption is the following:
6 F.BUCCI,D.TOUNDYKOV
Assumption 2.1 (Geometry of the domain). Γ is a level-surface of a convex
1
function and there exists x Rn such that
0
∈
(x x ) ν 0 on Γ ,
0 1
− · ≤
with ν being the outward normal vector field on Γ .
1
Figure 1 shows what a cross-sectionof a possible domain may look like.
χ
supp
x Ω
Γ Γ
0
1 0
Figure 1. A cross-sectionofa suitable domainΩ. The rigidwall
Γ , and, in particular, the part of the boundary outside suppχ
1
is convex; however, more general configurations are possible: for
other examples see [43].
Remark 2 (Moregeneralgeometricconditions). TheconvexityofΓ isasufficient
1
requirement, but it is possible to further relax this assumption. In particular, it
sufficestoaskforthesubsetofΓ thatliesawayfromsuppχ(x)tobealevel-surface
1
ofafunctionwithanon-vanishinggradientandpositive-definiteornegative-definite
Hessian(inthelattercasetheconditiononx shouldbechangedto(x x ) ν 0).
0 0
− · ≥
See [43] for more details and examples.
Assumption 2.2 (Nonlinear terms). The function f satisfies:
f C2(R) and there exists C such that
f
• ∈
(2.3) f′′(s) C (1+ s) s R;
f
| |≤ | | ∀ ∈
f(s)
limsup =: 2λ<0.
• s −
|s|→∞
Accordingly, set f (s)=f(s)+λs.
1
Existenceofaglobalattractorwillbeestablishedunderratherweakassumptions
on the feedback maps g and b.
Assumption 2.3 (Damping terms, I).
(1) g C(R)is amonotoneincreasingfunction suchthatg(0)=0; inaddition,
∈
g is linearly bounded at infinity, i.e. there exist positive constants m , M
g g
such that
(2.4) m s g(s) M s for s 1.
g g
| |≤| |≤ | | | |≥
(2) b C(R) is a monotone increasing function such that b(0) = 0; and there
∈
exists m such that
b
(2.5) b(s) m s for s 1.
b
| |≥ | | | |≥
ATTRACTORS FOR A COUPLED WAVE/PLATE SYSTEM 7
Afew commentsabout the aboveassumptionsare inorder. As observedalreadyin
[11] and [21], a prime consequence of Assumption 2.2 is local Lipschitz continuity
ofthe nonlineartermf (andoff , aswell),asanoperatorfromH1(Ω) intoL (Ω).
1 2
Namely, one has
f(ζ ) f(ζ ) C(r) ζ ζ ζ with ζ r, i=1,2.
1 2 1 2 1,Ω i i 1,Ω
k − k≤ k − k ∀ k k ≤
Theabovepropertyis easilyshownbyusingwellknownSobolevembeddingresults
in 3-D domains, which establish the critical threshold in the polynomial bound on
f. The same (Lipschitz continuity) is true for the nonlinear term f which occurrs
2
in the plate equation; more precisely,
f (w ) f (w ) C(r) w w w with w r, i=1,2.
k 2 1 − 2 2 k≤ k 1− 2k2,Γ0 ∀ i k ik2,Γ0 ≤
Next, observe that the basic Assumption 2.3 forces the acoustic damping to be
linearly bounded at infinity. It is important to emphasise that this condition is
actually necessary, because of the geometrical restrictions on the damping term.
This fact was first exhibited in [55] for a wave equation with boundary dissipation;
indeed,thecaseoflocaliseddissipationaroundaportionoftheboundaryyieldsthe
same technical difficulties and the same results as that of boundary damping. The
reader is referred to [21, Section 1] for a more detailed discussion of these issues.
The aforementioned basic assumptions are sufficient to establish both
well-posedness for the PDE problem (2.1) and existence of a global attractor for
the corresponding dynamical system.
Theorem 2.4 (Well-posedness). Define the state space
:=H1(Ω) L2(Ω) H2(Γ ) L2(Γ ).
Y × × 0 0 × 0
The system (2.1) generates a strongly continuous semigroup S(t) on . In partic-
Y
ular, given initial data, at t = 0, W = ζ ,ζ ,w ,w there exists a unique
0 0 1 0 1
{ } ∈ Y
solution to (2.1)
S()W = ζ,ζ ,w,w C([0,T], )
t t
· { }∈ Y
for any T 0. The solution satisfies the following variational identities
≥
T T T T
(ζ ,φ) (ζ ,φ )+ ( ζ, φ)+ (χg(ζ ),φ)
t t t t
− ∇ ∇
(2.6) (cid:12)0 Z0 Z0 Z0
(cid:12) T T
(cid:12)
(cid:12) = (f(ζ),φ)+ ακw ,φ ,
t
h i
Z0 Z0
T T T T
w ,ψ w ,ψ + ∆w,∆ψ + b(w ),ψ
t t t t
h i − h i h i h i
(2.7) (cid:12)0 Z0 Z0 Z0
(cid:12) T T
(cid:12)
(cid:12) = f (w),ψ ψ,βκζ ,
2 t
h i− h i
Z0 Z0
for any test functions
φ C1([0,T],L2(Ω)) C([0,T];H1(Ω)), ψ C1([0,T],L2(Γ )) C([0,T];H2(Γ )).
∈ ∩ ∈ 0 ∩ 0 0
Moreover, if the initial data belong to the spaces
ζ ,ζ H2(Ω) H1(Ω), w ,w H4(Γ ) H (Γ ) H2(Γ )
{ 0 1}∈ × { 0 1}∈ 0 ∩ 2 0 × 0 0
with the compatibility conditions (cid:0) (cid:1)
∂ζ ∂ζ
0 0
=0, =ακw ,
1
∂ν Γ1 ∂ν Γ0
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 F.BUCCI,D.TOUNDYKOV
then
ζ,ζ ,w,w L∞ 0,T;H2(Ω) H1(Ω) H4(Γ ) H2(Γ ) .
t t 0 0
{ }∈ × × ×
(cid:0) (cid:1)
We introduce the set ofequilibria for the dynamicalsystem ( ,S(t)); namely
N Y
(2.8) := W : S(t)W =W for all t 0 .
N { ∈Y ≥ }
Theorem 2.5 (Existence of the global attractor). Suppose that the Assumptions
2.1–2.2 hold. If the damping functions satisfy the Assumption 2.3, then the dy-
namical system ( ,S(t)) generated by the PDE problem (2.1) has a compact global
attractor A ,Ywhich coincides with the unstable manifold Mu( ) of the set
⊂Y N N
of stationary points:
A Mu( );
• ≡ N
lim dist(S(t)W, )=0 for any W .
• t→+∞ N ∈Y
Toprovethattheattractorhasafinitefractaldimensionwewillneedtostrengthen
the regularity and growth condition on both damping functions.
Assumption 2.6 (Dampings,II). Assume g,b C1(R), with g(0)=0=b(0), and
∈
(1) there exist positive constants m′, M′ such that
g g
(2.9) m′ g′(s) M′ s R,
g ≤| |≤ g ∀ ∈
(2) there exist positive constants m′, M′ such that
b b
(2.10) m′ b′(s) M′(1+sb(s)) s R.
b ≤| |≤ b ∀ ∈
Then, the following result holds.
Theorem 2.7 (Finite dimensionality and regularity of the attractor). Suppose the
hypotheses of Theorem 2.5 are satisfied. If, in addition, the Assumption 2.6 holds,
then the global attractor A has a finite fractal dimension.
Furthermore, the attractor A is bounded in the domain of the nonlinear semi-
group generator; in particular, A is a bounded subset of
H2(Ω) H1(Ω) H4(Γ ) H2(Γ ).
0 0
× × ×
2.2. Energy identity and bounds. With the system (2.1) we associate the fol-
lowing energy functionals:
1
(t) := ζ 2+λ ζ 2+ ζ 2 F (ζ(t))
ζ t 1
E 2 k∇ k k k k k −
ZΩ
1(cid:0) 1 (cid:1)
(t) := ∆w 2+ w 4+ w 2 Π(w(t)),
w t
E 2 k k 2k∇ k k k −
(cid:16) (cid:17)
where F (s) is the antiderivative of f (s) vanishing at 0, and
1 1
Q 1
(2.11) Π(w)=(p ,w)+ w 2 w 4.
0
2k∇ k − 4k∇ k
Define the total energy
(2.12) (t)= (t):=β (t)+α (t).
ζ,w ζ w
E E E E
The following identity satisfiedby (t) is standarddue to the fact thatthe dissipa-
E
tionfeedbacksarelinearlyboundedatinfinityandthesourcescorrespondtolocally
Lipschitz operators on the energy space. The equation can be derived for strong
solutions by using test functions βζ and αw in the variationalidentities (2.6) and
t t
ATTRACTORS FOR A COUPLED WAVE/PLATE SYSTEM 9
(2.7) respectively. Since the result is continuous with respect to the finite-energy
topology, it can be extended to all weak solutions:
t t
(2.13) (t)+β (χg(ζ ),ζ )+α (b(w ),w )= (s).
t t t t
E E
Zs Zs
Also, let us introduce positive quadratic energy functionals:
1 1 1
E (t):= ζ 2+λ ζ 2+ ζ 2 , E (t):= ∆w 2+ w 4+ w 2 ;
ζ t w t
2 k∇ k k k k k 2 k k 2k∇ k k k
(cid:0) (cid:1) (cid:16) (cid:17)
(2.14) E(t)=E (t):=E (t)+E (t).
ζ,w ζ w
Owing to the dissipativity property (in Assumption 2.2) satisfied by f and to
the structure of the functional Π in (2.11), it is not difficult to obtain upper and
lower bounds for the full energy of the system; see also [11, Section 2.2]. Explicit
computations pertaining to the wave component are found in [16, Section 2].
Proposition 2.8 (Bounds on the energy). Let be a bounded subset of . If
B Y
ζ ,ζ ,w ,w then there exists constants C ,C dependent only on the
0 1 0 1 1,B 2,B
{ } ∈ B
diameter of (in the topology of ) such that
B Y
(2.15) C E(t) C (t) (0) t 0.
1,B 2,B
− ≤E ≤E ∀ ≥
Weconcludethissectionbyintroducingtheabstractdynamicoperatorspertain-
ing to either equation, namely:
∂v
A v = ∆v, (A )= v H2(Ω) : =0 ;
1 1
− D ∈ ∂ν Γ
n (cid:12) o
A v =∆2v, (A )= v H4(Γ ) H1((cid:12)Γ ) :;∆v =0 .
2 D 2 ∈ 0 ∩ 0(cid:12) 0 ∂Γ0
n (cid:12) o
(cid:12)
3. The differences of trajectories: introductory results
Seeking to apply the abstract results recorded in the Appendix as Theorem A.2
andTheoremA.4,toinvestigatethe asymptoticbehaviourofthesolutionsto(2.1),
wemuststudydifferencesofitstrajectories,ratherthanthetrajectoriesthemselves.
In this section we introduce the relative basic definitions, along with a series of
preliminary identities which constitute a first step in the proof of our main results.
3.1. Auxiliaryfunctions and parameters. Giventwodifferentevolutiontrajec-
tories (h,h ,u,u ) and (ζ,ζ ,w,w ) of the coupled PDE system (2.1), we introduce
t t t t
the differences z and v, namely,
(3.1) z :=h ζ, (ζ =h+z);
−
(3.2) v :=u w, (u=w+v).
−
10 F.BUCCI,D.TOUNDYKOV
The pair (z,v) readily solves the new coupled system
z ∆z+λz+χ(x)g˜(ζ )=f˜(z) in Ω (0,T)
tt t 1
− ×
∂z
 =0 on Γ (0,T)
(3.3)  z∂∂∂(ννz0,=·)α=κhv0t−ζ0, zt(0,·)=h1−ζ1 oinn ΩΓ10××(0,T)
v +∆2v+˜b(w )+βκz =f˜(v) in Γ (0,T)
where we havevvt(st=0et,∆·)v==u00−w0t, vt(0t,|·Γ)0=u12−w1 ionn Γ∂Γ000.××(0,T)
(3.4) g˜(z ):=g(ζ +z ) g(ζ ),
t t t t
−
(3.5) ˜b(v ):=b(w +v ) b(w ),
t t t t
−
(3.6) f˜(z):=f (ζ+z) f (ζ),
1 1 1
−
(3.7) f˜(v):=f (w+v) f (w).
2 2 2
−
Technicallyeachoftheintroducedfunctionsalsodependsononeofthetermsinthe
corresponding difference, but that fact will be suppressed for brevity of notation.
3.2. Smoothcutofffunctions. Following[54,Section6],weintroducetwosmooth
cutoff functions whose role is to single out in Ω the dissipative and the non-
dissipative subdomains. More precisely, ψ,φ : Ω [0,1] are C2 functions with
→
the following properties:
supp(φ) supp(χ),
• ⊂
Ω supp(χ) x: ψ(x)=1 ,
• \ ⊂{ }
ψ 0 in a neighbourhood of Γ ,
0
• ≡
for any x Ω at least one of ψ(x) and φ(x) equals 1, namely
• ∈
x Ω:ψ(x)<1 φ−1( 1 ).
{ ∈ }⊂ { }
◦ ◦
By setting zˆ:=ψz and z :=φz, it is easily verified that zˆand z satisfy
(3.8) zˆ ∆zˆ+λzˆ+[[∆,M ]]z+ψχg˜(z )=ψf˜(z),
tt ψ t 1
−
(3.9) z◦ ∆z◦+λz◦ +[[∆,M ]]z+φχg˜(z )=φf˜(z),
tt φ t 1
−
whereM , M denote pointwise (a.e.) multiplicationby ψ orφ, respectively,while
ψ φ
the pairing [[, ]] denotes a commutator, acting as follows:
· ·
[[∆,M ]]z :=∆(ψz) ψ∆z.
ψ
−
A straightforwardcomputation gives the equivalent form
(3.10) [[∆,M ]]z =∆ψz+2 ψ z.
ψ
∇ ∇