wave equations under geometric conditions Local indirect stabilization of N–d system of two coupled C.R. Acad. Sci. Paris, Ser.I

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Partial differential equations/Optimal control

Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions

Stabilisation locale indirecte d’un système N–d de deux équations d’ondes couplées sous conditions géométriques

Chiraz Kassem

^a

^,

^c

, Amina Mortada

^b

, Layla Toufayli

^b

, Ali Wehbe

^b

aUniversitélibanaise,EDST&Hadath,Beyrouth,Liban

bUniversitélibanaise,Facultédessciences1etEDST&Hadath,Beyrouth,Liban

cUniversitéSavoieMontBlanc,LaboratoiredemathématiquesLAMA,bâtimentLeChablais,campusscientifique,73376LeBourget-du-Lac cedex,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received30December2017 Acceptedafterrevision7June2019 Availableonline27June2019 PresentedbyJean-MichelCoron

The purpose of this note is to investigate the stabilization of a system of two wave equations coupledby velocities with onlyone localized damping. The main novelty in this note is that the waves are not necessarily propagating at same speed and the couplingcoefficientis not assumedtobe positiveand small. Assume thatthe coupling regionand the damping region intersect. We prove that our system is strongly stable withoutgeometricconditions.Wethenstudytheenergydecayratebydistinguishingtwo cases.Thefirstoneiswhenthewaves propagateatthesamespeed.Inthiscase,under appropriategeometricconditions,weestablishanexponentialenergydecayestimatefor usualinitialdata.Fortheothercase,wefirstshowthatoursystemisnotuniformlystable.

Next,under the samegeometric conditions,weestablish apolynomial energydecayof type ¹_t forsmoothinitialdata.Finally,inonespacedimension,usingtherealpartofthe asymptoticexpansionofeigenvaluesofthesystem,weprovethattheobtainedpolynomial decayrateisoptimal.

©^{2019PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.}

ré s u m é

Nousnousintéressonsàlastabilisationd’unsystèmemultidimensionneldedeuxéquations d’ondescoupléesparlestermesdevitesse,etdontl’uneseulementestlocalementamortie.

Laprincipalenouveautécontenuedanscettenoteestquelesondes nesepropagentpas forcément àla même vitesse etque le coefficientde couplage n’est pas supposéêtre positif et petit. Nous supposons que la zone de couplage et la zone d’amortissement s’intersectent. D’abord, nous montrons que notre système est fortement stable sans conditions géométriques. Puis nous étudions le taux de décroissance de l’énergie en distinguant deux cas.Dansle premiercas, noussupposons que lesondes se propagent àlamêmevitesse ;nousétablissonsalors,souscertainesconditionsgéométriques,untaux dedécroissanceexponentieldel’énergiedusystèmepourdesdonnéesinitialesusuelles.

E-mailaddresses:[email protected](C. Kassem),[email protected](A. Mortada),[email protected](L. Toufayli), [email protected](A. Wehbe).

https://doi.org/10.1016/j.crma.2019.06.002

1631-073X/©^{2019PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.}

Danslesecondcas,nousmontronsd’abordquel’énergienedécroîtpasexponentiellement vers 0. Ensuite, sous les mêmes conditions géométriques, nous établissons un tauxde décroissancepolynomialdetype ¹_t pourdesdonnéesinitialesrégulières.Finalement,dans lecasparticulieroùladimensiondel’espaceestégaleà1,enutilisantlapartieréelledu développementasymptotiquedesvaleurspropresdesystème,nousmontrons,deplus,que letauxpolynomialobtenuestoptimal.

©^{2019PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.}

1. Introduction

Let

be a nonemptybounded open setof R^N having a boundary

of class C². In [8], F. Alabau-Boussouira et al.

consideredtheenergydecayofasystemoftwowaveequationscoupledbyvelocities:

u_tt

−

^a

u

+

^b

(

x

)

y_t

+ ρ (

x

,

u_t

) =

^{0 in}

× R

₊

,

(1.1)

y_tt

−

y

−

^b

(

x

)

u_t

=

^{0 in}

× R

₊

,

(1.2)

u

=

^y

=

^{0 on}

× R

₊

,

(1.3)

withthefollowinginitialdata:

u

(

x

,

0

) =

^u0

,

y

(

x

,

0

) =

^y0

,

u_t

(

x

,

0

) =

^u1andy_t

(

x

,

0

) =

^y

,

x

∈

wherea

>

0 isaconstantandb∈^C0

(;

R)isanon-zerofunction.Thedampingterm

ρ

isappliedtothefirstequationand thesecondequationisindirectlydampedthroughthecouplingbetweenthetwoequations.In[8],usinganapproachbased onmultipliertechniques,weightednonlinearinequalitiesandtheoptimal-weightconvexitymethod(developedin[5]),the authorsestablishedanexplicitenergydecayformulaintermsofthebehaviorofthenonlinearfeedbackclosetotheorigin.

Theirresultsareobtainedinthecasewhenthefollowingthreeconditionsaresatisfied:thewavespropagate atthesame speed(a=^{1),thecouplingcoefficientb}

(

x

)

issmallpositive(0≤^b

(

x

)

≤^b0,b0∈

(

0

,

b]^{wherebisaconstantdependingon}

andonthecontrolregion),andboththecouplingandthedampingregionssatisfyanappropriatedgeometricconditions namedPiecewiseMultipliersGeometric Conditions(introducedin[23],used in[5] anddenotedbyPMGC, inshort).Then the stabilizationof thesystem (1.1)–(1.3) in the casewherethe waves are not assumedto propagate withequalspeeds (aisnotnecessarilyequalto1)and/orwhenthecouplingcoefficientb

(

x

)

isnotassumedtobepositiveandsmallhasbeen left an open problemevenwhen thedamping term

ρ

is linearwithrespect tothe second variable.Inthisnote, we are interestedtoanswerthisopenquestionandtoprovideastabilityanalysisforthesystem(1.1)–(1.3) whenthedampingterm

ρ

islinearwithrespecttothesecond variablei.e.

ρ (

x

,

ut

)

=^c

(

x

)

ut wherec∈^C0

(

;R₊

)

.So,we considerthestability of thefollowingsystem:

u_tt

−

^a

u

+

^b

(

x

)

y_t

+

^c

(

x

)

u_t

=

^{0 in}

× R

₊

,

(1.4)

y_tt

−

y

−

^b

(

x

)

u_t

=

^{0 in}

× R

₊

,

(1.5)

u

=

^y

=

^{0 on}

× R

₊

,

(1.6)

withthefollowinginitialdata:

u

(

x

,

0

) =

^u0

,

y

(

x

,

0

) =

^y0

,

u_t

(

x

,

0

) =

^u1andy_t

(

x

,

0

) =

^y

,

x

∈ .

The notion ofindirect damping mechanismshas been introduced by D.L. Russell in [29], and since then, itattracted the attentionofmanyauthors. Inparticular, the stabilizationofsystems oftwo second-order equationscoupled through displacements when only one equation is effectively damped by internal or boundary feedback has been initiated and studiedin[1–3],andfurtherstudiedbymanyauthors,forinstance[6,24,17].Recallingthattheexponentialorpolynomial energydecayrateoccursinmanycontrolproblems,wequote[13,30] fortheTimoshenkosysteminboundedorunbounded domains. Here,we focusour attentiononlyon theliterature ofthe indirectinternal stability ofcoupled wave equations.

In [16], B. Kapitonov studied the stabilization of a systemof two coupled hyperbolic equationsinvolving (1.4)–(1.6). He established an exponential energydecay rateforusual initial data inthe casewherethe wavespropagated atthesame speed (a=^{1) andthedamping andcoupling coefficientshave thesamesupport.Forthe other cases,whena}=^{1 and/or} supportofbdoesnotcoincidewiththatofc,noenergydecayratehasbeendiscussed.In[2],F. Alabauetal.studiedthe indirectstabilizationofasystemoftwoevolutionequationscoupledthroughdisplacementswherethedampingiseffective inthewholedomain.Usingthemethodofhigher-orderenergiesinitiatedin[1],theyestablishedapolynomialenergydecay depending onthesmoothnessoftheinitialdata.TheseresultshavebeengeneralizedbyF.AlabauandM.Léautaud in[6]

to thecasewhen thecouplingandthedamping coefficientsarelocalizedin

andboth satisfythe PMGCconditions.In addition, without geometric conditions, using an interpolation inequality for ellipticsystem (see Proposition 5.1 in [18]) together withtheresolventestimatesofG.Lebeauin[19],theauthorsprovedthattheenergydecayofsmoothinitialdata is atleastlogarithmic whenthecouplingandthedampingregions intersectinanonemptysub-domain

ω

⊂

.However, when

ω

= ∅^{,thequestionofthestabilityorthenull}controllabilityofthesystemisstillanopenproblem.Indeed,F.Alabau andM.Léautaudin[7] solvedpartiallythisproblembyprovingthatthesystemisnullcontrollableprovidedthatboththe couplingandthedampingregions satisfytheoptimalgeometricconditionnamedGeometric ControlConditionintroduced by Bardos etal. in[11]. Finally,we referto [1,2,4,10,12,32,30,31,27,16,25,9] for theindirectstabilizationand theindirect exactcontrollabilityofdistributedsystemswithdifferentkindsofdamping.

Inthisnote,westudythestabilityofthesystem(1.4)–(1.6) whenthecouplinganddampingregionsintersectin

ω

⊂

. First, we establishthestrong stability withoutgeometric conditions.We thenstudythe energydecayrateofour system by distinguishing two cases.Thefirst oneis whenthewaves propagateatsame speed,i.e.a=^{1. Inthiscase, underthe} hypothesis that

ω

satisfies the geometric conditions PMGC(see below), we establish an exponential energy decay rate forusual initial data.Next,inthegeneralcase, whena=^{1, weprove the}non-uniform(exponential) stabilityand, under the samegeometric conditions,weestablish apolynomial energydecayrateoftype ¹_t forsmooth initial data.Finally,in one spacedimension,usingthe realpartoftheasymptoticexpansionoftheeigenvaluesofthesystem,we showthat the obtainedpolynomialdecayisoptimal.

2. Well-posednessandstrongstability

Inthissection,wewillstudythestrongstabilityofthesystem(1.4)–(1.6) withoutadditionalgeometricconditions.First, wewillstudytheexistence,uniqueness,andregularityofthesolutiontooursystem.

2.1. Well-posednessoftheproblem

LetU=

(

u

,

u_t

,

y

,

y_t

)

bearegularsolutionto(1.4)–(1.6),itsassociatedenergyisdefinedby E

(

t

) =

¹

2

|

^ut

|

²

+

^a

|∇

^u

|

²

+ |

^yt

|

²

+ |∇

^y

|

²

dx

.

(2.1)

So,adirectcomputationgives d

dtE

(

t

) = −

c

(

x

) |

^ut

|

^2dx

≤

⁰

.

(2.2)

Consequently,thesystem(1.4)–(1.6) isdissipativeinthesensethatitsenergyisnon-increasing.

First,we definetheenergyspaceH=

(

H₀¹

()

×^L2

())

² equipped,forallU=

(

u

,

v

,

y

,

z

),

U=

(

^u

,

^v

,

^y

,

^z

)

∈H^,bythe scalarproduct:

(

U

,

U

)

_H

=

^a

(∇

^u

· ∇

^u

)

dx

+

v

^vdx

+

(∇

^y

· ∇

^y

)

dx

+

z

^zdx

.

Next,wedefinetheunboundedlinearoperatorA:^D

(A)

→H^by:

D

(

A

) = (

H²

() ∩

^H10

() ×

^H10

())

²

,

AU

= (

v

,

a

u

−

^bz

−

^cv

,

z

,

y

+

^bv

).

Notethat, usingthefact thatc

(

x

)

≥^0,then A^ism-dissipativeandgeneratesa C0 semi-groupofcontractionse^tA onthe energyspaceH^.Asthesystem(1.4)–(1.6) isequivalentto

U_t

=

A^UinH

,

t

>

0

,

U

(

0

) =

^U0 (2.3)

withU=

(

u

,

ut

,

y

,

yt

)

,wededuceitswell-posedcharacter.So,wehavethefollowingexistenceresults:

Theorem2.1.LetU₀∈H^{then,problem(2.3)admitsauniqueweaksolution}U satisfying U

(

t

) ∈

^C0

R

⁺

,

H

.

Moreover,ifU0∈^D

(A)

^{then,problem(2.3)admitsauniquestrongsolution}U satisfying U

(

t

) ∈

^C1

R

⁺

,

H

∩

^C0

(R

⁺

,

D

(

A

)).

Secondly,wewillstudythestrongstabilityofoursystem.

2.2. Strongstability

Inthissubsection,westudytheasymptoticbehaviorofE

(

t

)

.Forthisaim,weassumethatthereexistsanonemptyopen

ω

c₊⊂

satisfyingthefollowingcondition:

{

^x

∈ :

^c

(

x

) >

0

} ⊃ ω

c₊ (LH1)

Ontheotherside,asb

(

x

)

isanon-zerocontinuousfunction,thenthereexistsanonemptyopen

ω

b₊∪

ω

b₋⊂

suchthat

{

^x

∈ :

^b

(

x

) >

0

} ⊃ ω

_b+ and

{

^x

∈ :

^b

(

x

) <

0

} ⊃ ω

_b− (LH2) Ourmainresultinthispartisthefollowing.

Theorem2.2(Strongstability).Assumethata

>

0,thatcondition(LH1)holdsandthat

ω

=

ω

c₊∩

ω

b₊= ∅^or

ω

c₊∩

ω

b₋= ∅^.Then thesemigroupofcontractionse^tAisstronglystableontheenergyspaceH^{,i.e.,foranyU}0∈H^,wehave

t→+∞lim

^etAU0

H

=

⁰

.

(2.4)

In[6], theauthors consideredthe stabilizationofasystemoftwo wave equationscoupledindisplacements withone localized internal damping. They showed that, under the assumption that the damping region and the coupling region have a non-empty intersectionin

,i.e.

ω

₌

ω

c₊∩

ω

b₊= ∅ ^(or

ω

c₊∩

ω

b₋= ∅^{), the energy ofsmooth solutions decays} logarithmically tozeroast goesto infinity.Thisresultstill holdsinthe casewherethetwo waveequations arecoupled throughthevelocities.Indeed,followingthemethodintroducedbyG.LebeauandL.Robbianoin[20],M.Léautaudin[18]

establishedaninterpolationinequalityfortheassociatedellipticsystem. Thisinterpolationinequalityimpliestheresolvent estimatesof G. Lebeauin [19] (see also[21]) that providethe logarithmic energydecay rateforsmooth initial data.So, usingthedensityof D

(A)

inHandthecontractionpropertyoftheC0 semigroupe^tA,wededucethattheenergyofthe system(1.4)–(1.6) decaysasymptoticallytozeroast goestoinfinityforallusualinitialdata.

Thenweareinterested,inthispaper,tostudytheenergydecayratebydistinguishingtwocases.

3. Exponentialstability,thecasea=¹

Thissectionisdevotedtothestudyoftheexponentialstabilityofsystem(1.4)–(1.6) incasethewavespropagateatthe same speed(in the casea=^{1) andunder}appropriatedgeometric conditions. Forthat purpose,we willuse a frequency domainapproachcombinedwithapiecewisemultipliertechnique.

Beforepresentingourmainresultofthissection,werecallthepiecewisemultiplier geometricconditionintroduced by K.Liuin[23].

Definition3.1.Wesaythat

ω

satisfiesthepiecewisemultiplier geometriccondition(PMGCinshort)ifthereexist

j⊂

havingLipschitzboundary

j=

∂

jandxj∈R^N, j=¹

, ...,

Jsuchthat

j∩

i= ∅^{for j}=^iand

ω

containsaneighborhood in

oftheset∪_j^J₌₁

γ

j

x_j

∪

\ ∪_j^J₌₁

j

where

γ

j

(

x_j

)

= {^x∈

j:

(

x−^xj

)

·

ν

j

(

x

) >

0}^and

ν

jistheoutward unitnormal vectorto

j.

Remark3.2.ThePMGCisthegeneralization ofthe multipliersgeometric condition(MGC inshort) introducedbyLionsin [22],sayingthat

ω

containsaneighborhood in

oftheset{^x∈

:

(

x−^x0

)

·

ν (

x

) >

0}^{,forsome x}0∈R^N,where

ν

isthe outwardunitnormalvectorto

=

∂

.

Now,weareinapositiontopresentthemainresultofthissection.

Theorem3.3(Exponentialdecayrate).Leta=^{1.Assumethatcondition(LH1)holds.Assumealsothatthenonemptyopenset}

ω

=

ω

c₊∩

ω

b₊(or

ω

=

ω

c₊∩

ω

b₋)satisfiesthegeometricconditionsPMGCandthatb

,

c∈^W1,∞

()

.Thenthereexistpositiveconstants M≥^1,

θ >

0suchthat,forallinitialdata

(

u₀

,

u₁

,

y₀

,

y₁

)

∈H^{,theenergyofthesystem}(1.4)–(1.6) satisfiesthefollowingdecayrate:

E

(

t

) ≤

^Me−θtE

(

0

), ∀

^t

>

0

.

(3.1)

Remark3.4.Notethat inTheorem 3.3,we haveno restriction on theupper bound andthe signof the function b.This theorem isthen a generalization in the linearcaseof the resultof [8] wherethe coupling coefficient considered hasto satisfy 0≤^b

(

x

)

≤^b0, b₀∈

(

0

,

b] ^{where b is a constant depending on}

and on the control region. Nevertheless, the problemisstillopeninthenonlinearcase.

Inordertoprovetheabovetheorem,weapplyaresultofF.L.Huang[15] andJ.Pruss[28]:aC0-semigroupofcontraction

(

e^tA

)

t0 inaHilbertspaceH^{isuniformlystableifandonlyif}

i

R ⊆ ρ (

A

)

(H1)

and sup

β∈R

(

i

β

I

−

A

)

⁻¹

< +∞

^(H2)

hold.

Since theresolventofA^{iscompactand0}∈

ρ (

A

)

,then fromTheorem2.2,wededucethat condition(H1) issatisfied.

We nowprove that condition(H2) holds,usingan argumentofcontradiction. Tothisaim,wesuppose that thereexista realsequence

β

nwith

β

n→ +∞^{andasequenceU}n=

(

un

,

vn

,

yn

,

zn

)

∈^D

(

A

)

suchthat

(

un

,

vn

,

yn

,

zn

)

_H

=

1

,

(3.2)

and

nlim→∞

(

i

β

nI

−

A

)

Un

_H

=

0

.

(3.3)

Now,detailingEq. (3.3),weget

i

β

nun

−

^vn

=

^fn¹

→

^{0 in H1}0

(),

(3.4)

i

β

nvn

−

un

+

^b

(

x

)

zn

+

^c

(

x

)

vn

=

^g1n

→

^{0 in L2}

(),

(3.5)

i

β

nyn

−

^zn

=

^fn²

→

^{0 in H1}0

(),

(3.6)

i

β

nzn

−

yn

−

^b

(

x

)

vn

=

^g2n

→

^{0 in L2}

().

(3.7)

Eliminating v_n andz_n fromtheprevioussystem,weobtainthefollowingreducedsystem

β

_n²un

+

un

−

ⁱ

β

nb

(

x

)

yn

−

ⁱ

β

nc

(

x

)

un

= −

^gn¹

−

^b

(

x

)

f_n²

−

ⁱ

β

nf_n¹

−

^c

(

x

)

f_n¹

,

(3.8)

β

_n²y_n

+

y_n

+

ⁱ

β

nb

(

x

)

u_n

= −

ⁱ

β

nf_n²

+

^b

(

x

)

f_n¹

−

^gn²

.

(3.9) Ontheotherside,usingEq. (3.2) wededucethat z_n andv_n areuniformlyboundedinL²

()

.Itfollows,fromEqs. (3.4) and (3.6),that

|

^yn

|

^2dx

=

^O

(

1

) β

_n^{2 and}

|

^un

|

^2dx

=

^O

(

1

)

β

_n²

.

(3.10)

Lemma3.5.Thesolution

(

un

,

vn

,

yn

,

zn

)

∈^D

(

A

)

tothesystem(3.4)–(3.7) satisfiesthefollowingestimation

ωc+

|β

nu_n

|

^2dx

=

^o

(

1

).

(3.11)

Proof. First,sinceUnisuniformlyboundedinH^{,thenfrom(3.3),weget} Re

i

β

n

^Un

²

−(

A^Un

,

U_n

) =

c

(

x

)|

^vn

|

^2dx

=

^o

(

1

).

(3.12)

Undercondition(LH1),itfollowsthat

ωc+

|

^vn

|

^2dx

=

^o

(

1

).

(3.13)

So,usingEqs. (3.12) and(3.4),weget

c

(

x

) | β

_nu_n

|

^2dx

=

^o

(

1

).

(3.14)

Consequently,wehave

ωc+

|β

nu_n

|

^2dx

=

^o

(

1

).

Theproofisthuscomplete. 2

Now,thesubset

ω

satisfiesthePMGC.Hence, denotingby

j andx_j, j=¹

, ...,

J thesetsandthepointsgivenbythe PMGC,wehave

ω

⊃N

∪^J_j=1

γ

j

xj

∪

\ ∪^J_j=1

j

∩

.Inthisexpression,N

(

O

)

= {^x∈R^N:^d

(

x

,

O

) < ε

_}withd

(

·

,

O

)

istheusual Euclideandistancetothesubset O ofR^N and

γ

j

(

xj

)

= {^x∈

j:

(

x−^xj

)

·

ν

j

(

x

) >

0}^where

ν

j istheoutward unitnormalvectorto

j=

∂

j.Letthereals0

< ε

1

< ε

2

< ε

3

< ε

anddefine

Vi

=

Nεi

∪

^J_j=1

γ

j

xj

∪

\ ∪

_j^J₌₁

j

,

i

=

1

,

2

,

3

.

Since

\V3

∩V2= ∅^{,thenwemaydefinethefunction}

η

∈^C∞₀

(R

^N

)

by

η (

x

) =

^{0 if x}

∈ \

V3

,

0

≤ η (

x

) ≤

¹

, η (

x

) =

^{1 if x}

∈

V2

.

Lemma3.6.Thesolution

(

un

,

vn

,

yn

,

zn

)

∈^D

(

A

)

tothesystem(3.4)–(3.7) satisfiesthefollowingestimation

η (

x

) |∇

un

|

²dx

=

^o

(

1

)

and

V2∩

|∇

un

|

²dx

=

^o

(

1

).

(3.15)

Proof. MultiplyingEq. (3.8) by

η

u¯nandusingGreen’sformulaandthefactthatun=^{0 on}

,weobtain

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

η (

x

) | β

nun

|

^2dx

−

η (

x

) | ∇

^un

|

^2dx

−

un

( ∇ η _{· ∇}

un

)

dx

−

ⁱ

β

n

b

(

x

) η

ynundx

−

ⁱ

β

_n

c

(

x

) η (

x

) |

^un

|

^2dx

=

( −

^g1n

−

^b

(

x

)

f_n²

−

ⁱ

β

_nf_n¹

−

^c

(

x

)

f_n¹

) η

u_ndx

.

(3.16)

As f_n¹ and f_n² convergetozeroinH¹₀

()

,g¹_n convergestozeroinL²

()

,thesequences

(β

nun

)

,

(β

nyn

)

,

(

∇^un

)

areuniformly boundedinL²

()

and^un=^o

(

1

)

,weget

η (

x

) | β

nun

|

^2dx

−

η (

x

) | ∇

^un

|

^2dx

=

^o

(

1

).

(3.17)

Byusingthedefinitionof

η

andEqs. (3.11) and(3.17),wededuce

η (

x

) | ∇

^un

|

^2dx

=

^o

(

1

)

and

V2∩

| ∇

^un

|

^2dx

=

^o

(

1

).

Theproofisthuscomplete. 2

Lemma3.7.Thesolution

(

un

,

vn

,

yn

,

zn

)

∈^D

(

A

)

tothesystem(3.4)–(3.7) satisfiesthefollowingestimation

η (

x

) |∇

^yn

|

^2dx

=

^o

(

1

)

and

V2∩

|∇

^yn

|

^2dx

=

^o

(

1

).

(3.18)

Proof. Theproofcontainsthreepoints.

(i) Notice that, from equation (3.9), _β¹

n

¯

^yn is uniformly bounded in L²

()

. So, multiplying Eq. (3.8) by _β¹

n

η ¯

^yn. Using Green’sformulaandthefactthatun=^yn=^fn¹=^{0 on}

,weget

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

−

β

n

η (

x

)(∇

^yn

· ∇

^un

)

dx

−

β

nu_n

(∇ η · ∇

^yn

)

dx

+

1

β

n

η (

x

)

y_n

u_ndx

+

i

b

(

x

) η (

x

) | ∇

yn

|

²dx

+

i

η (

x

)

yn

(∇

b

· ∇

y_n

)

dx

+

i

b

(

x

)

yn

(∇ η · ∇

y_n

)

dx

+

ⁱ

c

(

x

) η (

x

)( ∇

^un

· ∇

^yn

)

dx

+

ⁱ

c

(

x

)

un

( ∇ η _{· ∇}

y_n

)

dx

+

ⁱ

η (

x

)

un

( ∇

^c

· ∇

^yn

)

dx

=

+

( −

^gn¹

−

^b

(

x

)

f_n²

−

^c

(

x

)

f_n¹

)(

¹

β

_n

η (

x

)

y_n

)

dx

+

ⁱ

η (

x

)( ∇

^fn¹

· ∇

^yn

)

dx

+

ⁱ

f_n¹

(∇ η · ∇

^yn

)

dx

.

(3.19)

First,since f_n¹, f_n² convergetozeroinH₀¹

()

,g_n¹ convergestozeroinL²

()

and

(

∇^yn

)

,

(

_β¹

n

yn

)

areuniformlyboundedin L²

()

,thenwehave

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

(−

^g_n¹

−

^bf_n²

−

^{c f}_n¹

)(

¹

β

n

η

y_n

)

dx

+

ⁱ

η (∇

^f_n¹

· ∇

^yn

)

dx

+

ⁱ

f_n¹

(∇ η · ∇

^yn

)

dx

=

^o

(

1

).

(3.20)

Next,usingthedefinitionof

η

,theEqs. (3.11),(3.15), andthefact that^un=^o

(

1

)

,^yn=^o

(

1

)

and

(

∇^yn

)

isuniformly boundedL²

()

,weget

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

−

β

_nu_n

( ∇ η · ∇

^yn

)

dx

+

ⁱ

η (

x

)

y_n

( ∇

^b

· ∇

^yn

)

dx

+

ⁱ

b

(

x

)

y_n

(∇ η · ∇

^yn

)

dx

+

ⁱ

c

(

x

) η (

x

)(∇

^un

· ∇

^yn

)

dx

+

ⁱ

c

(

x

)

u_n

(∇ η · ∇

^yn

)

dx

+

ⁱ

η (

x

)

u_n

(∇

^c

· ∇

^yn

)

dx

=

^o

(

1

).

(3.21)

Finally,inserting(3.20) and(3.21) into(3.19),weget

−

β

n

η (∇

^yn

· ∇

^un

)

dx

+

1

β

n

η

y_n

u_ndx

+

ⁱ

b

η | ∇

^yn

|

^2dx

=

^o

(

1

).

(3.22)

(ii)Multiplying(3.9) bytheboundedsequence _β¹

n

η

un,integratingover

andusingthefactthat un=^yn= ^fn²=^{0 on}

, weget

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

−

β

n

η (

x

)(∇

yn

· ∇

un

)

dx

− β

n

yn

(∇ η · ∇

un

)

dx

+

1

β

n

η (

x

)

yn

undx

−

ⁱ

η (

x

)

un

( ∇

^b

· ∇

^un

)

dx

−

ⁱ

η (

x

)

b

(

x

) |∇

^un

|

^2dx

−

ⁱ

b

(

x

)

un

( ∇

^un

· ∇ η )

dx

=

i

η (

x

)

∇

^fn²

· ∇

^un

+

^fn²

∇ η _{· ∇}

un

dx

+

b

(

x

)

f_n¹

−

^gn²

₁

β

n

η (

x

)

undx

.

(3.23)

First,since f_n¹, f_n² convergetozeroin H¹₀

()

, g²_n convergesto zeroinL²

()

and∇^un, _β¹

n

un are uniformlyboundedin L²

()

,thenwehave

i

η (

x

)

∇

^fn²

· ∇

^un

+

^fn²

∇ η · ∇

^un

dx

+

b

(

x

)

f_n¹

−

^gn²

₁

β

n

η (

x

)

u_ndx

=

^o

(

1

).

(3.24)

Next,usingEq. (3.15) andthefactthat^un=^o

(

1

)

and

(β

ny_n

)

isuniformlyboundedinL²

()

,wededucethat

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

−β

n

yn

(∇ η · ∇

un

)

dx

−

i

η (

x

)

un

(∇

b

· ∇

un

)

dx

−

ⁱ

η (

x

)

b

(

x

) |∇

^un

|

^2dx

−

ⁱ

b

(

x

)

un

( ∇

^un

· ∇ η )

dx

=

^o

(

1

).

(3.25)

Finally,inserting(3.24) and(3.25) into(3.23),weget

−

β

n

η (

x

)( ∇

^yn

· ∇

^un

)

dx

+

1

β

n

η (

x

)

yn

undx

=

^o

(

1

).

(3.26)

(iii)SummingtheimaginarypartsofEqs. (3.22) and(3.26),weobtain

b

(

x

) η (

x

) |∇

^yn

|

^2dx

=

^o

(

1

).

(3.27)

Usingthedefinitionofthefunction

η

andcondition(LH2),wededucethat

η (

x

)|∇

^yn

|

^2dx

=

^o

(

1

)

and

V2∩

|∇

^yn

|

^2dx

=

^o

(

1

).

Theproofisthuscomplete. 2

Lemma3.8.Thesolution

(

un

,

vn

,

yn

,

zn

)

∈^D

(

A

)

tothesystem(3.4)-(3.7) satisfiesthefollowingestimation

η (

x

) | β

nyn

|

^2dx

=

^o

(

1

)

and

V2∩

| β

nyn

|

^2dx

=

^o

(

1

).

(3.28)

Proof. MultiplyingEq. (3.9) by

η

¯yn andintegratingover

.Then,usingGreen’sformulaandthefactthat^yn=^o

(

1

)

and y_n=^{0 on}

,weget

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

η (

x

)|β

nyn

|

²dx

−

y_n

(∇

yn

· ∇ η )

dx

−

η (

x

) | ∇

yn

|

²dx

+

i

β

n

b

(

x

) η (

x

)

ynundx

=

o

(

1

).

(3.29)

CombiningEqs. (3.11),(3.18) and(3.29) andusingthefactthat^yn=^o

(

1

)

,weget

η (

x

) | β

nyn

|

^2dx

=

^o

(

1

)

and

V2∩

| β

nyn

|

^2dx

=

^o

(

1

).

Theproofisthuscomplete. 2

Now,since

(

j\V2

)

∩V1= ∅^{,wecandefinethefunction}

ψ

j∈^C₀^∞

(R

^N

)

by:

ψ

j

(

x

) =

0 ifx

∈

V1

,

0

ψ

j

1

, ψ

j

(

x

) =

1 ifx

∈

j

\

V2

.

Formj

(

x

)

=

(

x−^xj

)

,wedefinehj

(

x

)

=

ψ

j

(

x

)

mj

(

x

)

.