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Stochastic Wave Equations with Memory

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Stochastic Wave Equations with Memorywave,with,Wave,WAVE,WITH

    Stochastic Wave Equations with Memory C_.Ann.Math.

    alB(3),2010,329342

    DOI:10.1007/sl14010090170X

    ChineseAnnalsof

    Mathematics,SeriesB

    ?TheEditorialOfficeofCAMand

    SpringerVerlagBerlinHeidelberg2010

    StochasticWaveEquationswithMemory

    TingtingWEIYimingJIANG

    AbstractTheauthorsshowtheexistence

    chasticwaveequationswithmemory.The

    solutionisobtainedaswel1.

    anduniquenessofsolutionforaclassofsto-

    decayestimateoftheenergyfunctionofthe

    KeywordsStochasticwaveequationswithmemory,Resolvent,Infinitedimensional Wienerprocess,Energyfunction

    2000MRSubjectClassificati0n60H1535L05

    1Introduction

    Thewaveequationwithmemoryofthefollowingform

    futt--/kU+(ts)diV{.()u}ds+,()+6(z)h()=.{(,)=0,onOD,

    l(0):UO,

    Ut(0)=u1,(z,t)?D×[0,T

    describesthemodelofmaterialsconsistingofanelasticpartfwithoutmemory)andaviseoelas

    ticpart(memory)withu(t,X)givingthepositionofmaterialparticleattimet.Theterm iStherelaxationfunction./denotesthebodyforceandhisthedampingterm.Theproperties

    ofthesolutiontof1.1)hasbeenstudiedbymanyauthors.Forthecasethatthedampingterm iSzero,Dafermosf101provedthatthesolutionoftheviscoelasticsystemdecaystozeroastime

    goestoinfinity.Unfortunately,theexplicitratewasnotobtained.RiveraI14Iobtainedthe unifornlratesofdecayforthesolutionofalinearviscoelasticsystemwithmemory,basedon second

    orderestimates.Forthepartiallyviscoelasticcase,RiveraandSalvatierraI13lshowed thattheenergyofthesolutiondecaysexponentiallywhen8decaysexponentially.Ontheother hand,Cavalcantiand0quendof61studiedthenonlinearequationwithanonlinearandlocal- izedfrictionaldampingandprovedtheexponentialandpolynomialdecayratesoftheenergy. Recently,Alalau

    Boussouiraetalf11developedaunifiedmethodforthedecayestimatesofthe energyofthegeneralsecondorderintegrodifferentialequations.

    Infact,thedrivingforcemaybeaffectedbytheenvironmentrandomly.Inviewofthis,we ManuscriptreceivedMay12,2009.RevisedOctober9,2009.PublishedonlineApril20,2010 SchoolofMathematics,NankaiUniversity,Tianjin300071,China.

    Email:swindway@mail.nankai.edu.caymjiangnk@nankM.edu.cn

    ProjectsupportedbytheNationalNaturalScienceFoundationofChina(No.10871103). 33O

    considerthefollowingstochasticwaveequationwithmemory

    u

    ;tt-Au+,2a.Unta+.f,O(?一s)?乱ds+,().T.WeiandM.Jiangw(t,z),

    (1.2)

    for(x,t)?Dx0,'where?denotestheLaplaceoperatoronDwithDirichletboundary

    condition,0isaconstant,andWisaninfinitedimensiona1Wienerprocesswhichmaybe treatedastherandomforce.DCisaboundedopendomainwithsomesmoothboundary aDford>3,andT>0isaconstant.

    Beforesolvingtheequationabove,wefirstmentionsomeimportantstudiesonthegeneral stochasticwaveequations.Usingestimatesontheenergyfunction,Chow[7]provedtheexis

    tenceofaglobalsolutiontostochasticwaveequationswithapolynomia1nonlinearity.With thesimilarmethod,Chow8,9Istudiedfurtherpropertiesofthesolutionsuchasasymptotic stabilityandinvariantmeasure.Ontheotherhand,Barbuetalf21obtainedtheexistence anduniquenessoftheinvariantmeasureofthesolutionwithoutthecomputationoftheen- ergy.Moreover,usingtheenergyinequality,Boetal[3]proposedsufficientconditionsthatthe solutionsofaclassofstochasticwaveequationsblowupwithapositiveprobabilityorinL sense.However,forthecurrentequation(1.2),thememorypartmakesitdimculttoestimate theenergybyusingthesemethods.Hence,wesolveitinanotherway.Wleusethedefinition ofsolutionsinf41andextendthemtothestochasticcases.Thenwefirstprovetheexistence anduniquenessofalocalmildsolution.Followingfromtheargumentsforthedecayestimate ontheenergyfunctionin[116,13,14],weprovetheglobalexistenceofthesolution. Theremainingpartofthisarticleisorganizedasfollows.Definitionsofthesolutionto Equation(1.21aregiveninSection2.InSection3,weshowthelocalexistenceanduniqueness ofthemildsolution.InSection4,thedecayestimateoftheenergyfunctionisobtainedand theglobalexistenceofthesolutionisproved.

    2Preliminaries

    Accordingtotheargumentsin[12and[4jwesolveequation(1.1)asanintegro

    differential

    equation.Moreprecisely,considerthefollowingintegral-differentialequation fB(t?)=0(2.1)withu?

    L([0,;),whereXisarealHilbertspace,AandBsatisfytheconditionsin

    ieAandB(?)arelinearunboundedself-adjointoperatorswithdomainsD(A)andD(B( respectively,satisfyingthat

    (A1)D(A)cD(B(t))foranyt0andD(A)isdenseinX.

    (A2)(Ay,Y)a011ullforanyY?D(A)andsomeconstanta0>0

    (A3)B(.)?.11(0,+?;)foranyY?D().

    ;

    StochasticWaveEquationswithMemo~

    (A4)B(t)commuteswithA,thatis

B(t)D(A)cD(A)andAB(t)y=B(t)Ay,Y?D(A.),t0

    331

    Here,fbecomestheoperatordefinedby,(@))()=f(u(t,)),andthesimilarchangeis madeonh.

    Definition2.1Afamilyboundedlinearoperators{s(?)}t0iniscalledaresolvent

    ,0requation(2.1)withf=0andh=0,ifthefollowingconditionsaresatisfied: (s1)s(o)=Iands(t)isstrongcontinuouson[0,..).Thatis,forallX?X,S(?)Xis

    continuouson0,..).

    (s2)s(t)commutesA,whichmeansthatS(t)D(A)cD(A)andAS(t)y=S(t)Ayfornff Y?D(A)andt0.

    (s3)F0ranyY?D(),S(?)YistwicecontinuouslydifferentiableinXon[0,o.)and s(o)=0.

    (s4)ForanyY?D(A)andt0,theresolventequationis

    (?)y+AS(t)+tB(

    r)s)dr=.

    ItiseasytocheckthatwhenA=一?and

    onDwithDirichletboundaryconditionand

    t,AandBsatisfytheconditions(A1)(A4).

    (2.2)

    B(t)=9(t)A,with?beingtheLaplaceoperator

    (?)beingcontinuousdifferentiablefunctionon

    Let(.,.)betheinnerproductandl1.1lbethenormontheHilbertspaceL(D),andwe havethefollowingtheorem,whichisaconsequenceofTheorem2in[4]. Theorem2.1AssumethatA=-AandB(t)=Z(t)A,with?beingtheLaplaceopera

    toronDwithDirichletboundaryconditionand(?)beingcontinuousdifferentiablefunction

    ont,f=0andh=0.Thenthereexistsauniqueresolvent{S(?))t0dorequation(2.1).

    Furthermore,theresolventsatisfiesthefollowingproperties: (i)Theoperatorss(t)areself-adjoint.

    (ii)s()commuteswith4-A,thatisS(t)D(v/-A)CD(,//)and,/s(t)x=(?),/A

rall?D(,/A)andt0.

    (iii)Forany?L.(D),efunctiontS(r)xdrbelongstoc(0,..);D(,/))and

    ranyT>0,thereexistsaconstantCTsuchthat s(t)zll+~f0's(r)drIIlIII,0r.n?[.,](2.3)

    (iv)Forany?D(v/-A),thefunctiontS(r)xdrbelongstoc([0,oc);D(v/-A))and {oranyT>0

    A/ots(r)drII<IIIlj??【.,

    llllcr(1lxll+IIv~xl1),t?【0,T]

    Sx+A]ots(r)xdr+B,1,S(t)=.,t.,

    wherestandsfortheconvolutionoftwofunctions. (v)Forany?D()thefunction(.xbelongstoc([0,?);D(v/-A))

    (2.4)

    (2.5)

    (2.6)

    332.

    T.WeiandM.Jiang

    Withtheresolvent,wecandefinethesolutionsofthestochasticwaveequation(1.2).We

    rewritetheequation?LS

    州?一?,

    whereA:A,B(t)

    (2.7)

    ()?,andWisaQWienerprocessinXonsomeprobabilityspace ,)withthevarianceoperatorQsatisfyingnQ<..and{,t0}asitsnatural

    filtrationsatisfyingtheusualconditions.(Here,weuseformdw(t)insteadof(?)todenote thewhitenoiseintimesincewewillusetheIt6formulawithrespecttotheinfinitedimensional

    Wienerprocess.)Moreover,wecanassumethatQhasthefollowingform

    Qei:九?i,i=1,2

    where{)areeigenvaluesofQsatisfying?i<(20andei)arethecorrespondingeigen

    functionswhichformanorthonoa1baseofX

    .Inthiscase.

()=?(?)e,

    i=1

    where{()}isasequenceofindependentcopiesofstandardBrownianmotionsin0nedimen sion.Let7-/bethesetofLo=L.(Qi1,).valuedprocesseswiththenorm

    2dsJ

    .n((?(s)Q)((s)Q))ds]<..

    where((s)Q{)denotestheadjointoperatorof(s)Q.Let{t):1beapartitionon[0, suchthat0=to<tl<<tn:T.

    Foraprocess?.definethestochasticintegralwith respecttotheQWienerprocessas

    (s)dW(s)札一l=lim

    n}?—

    k=O

    (%)((t+1At)W(tkA?))

    wh

    .

    erethesequenceconvergesinsense.

    Itisnotdifficulttocheckthattheintegralprocess 9(8)dW(s)isamartingaleforany?7-/,andthequadraticvariationpr0cessjsgivenby

    )d(S)=(()1()s

    Inparticular,ifwetake1.thentheequationabovebecomes "(?)))=/0n((Qi1J(1))ds=tTrQ

    FormoredetailsabouttheinfinitedimensionWienerprocessandthestochasticintegra1

    ,we

    referto111.

    ::

    StochasticWaveEquationswithMemo~ Definition2.2Letf?(0,T]×L(J[));L.(D)).

    (i)Wesaythatisnstrongsolutionto(1.2)

C([0,T]×Q;X)n([0,T】×Q;D())andsatisfies(1.2)

    is<)oadaptedthatbelongsto (ii)An{)t>oadaptedXvaluedstochasticprocessissaidtoben

    u?C([0,T】×Q;X)nc([o,T】×Ft;D(,/)),andforany?D(v~A), holds:

    weaksolutionto(1.2)

    thefoUowingequation

    ()+(()//os),"(s)ds,)

    -

    2a(u,,)(,,)+(,dw(?))

    (iii)An{)t?0adaptedXvaluedstochasticprocessissaidtobeamildsolutionto

    ifu?C([0,】×Q;)n([0,】×Q;D())andthefollowingequationholds:

    u(t)=).+t)

    

    JU

    

    2Qls(tr)udr0

    

    t

    l*S(?一r),(r,"(r))d

    =

    )u0+2at)

    .dr+

    

    

    /1

    o

    r+

    t

_r)d?

    s(r)Uldr-2aJ0s(r)u(r)dr

    (tr),("(r))dr+t1

    *S(?一r)d(r)

    whereSistheresolveoperatorof(2.1)withf=0andg=0

    (2.8)

    (1.2)

    (2.9)

    Remark2.1BvthedefinitionsaboveandTheorem2.1,wehavethefollowingfacts. fi1Thestochasticintegralin(2.9)iSwell-defined.

    fii1Astrongsolutionof(1.2)isalsoaweaksolutionandamildsolution. fiii1Ifisamildsolutionto(1.2)satisfyingU0?D(A)andUl?D(4A),thenUisa

    strongsolution.

    fiv1Aweaksolutionmaybenotequivalenttoamildsolution,becausetheresolveGmay benotasemigroup.WhentheoperatorBin(2.7)equals0,(2.7)degeneratestothegeneral stochasticwaveequationandthemildsolutionisequivalenttotheweaksolutionthen. Here.wefocuSonthemildsolutiontoequation(1.2)andgivetheexistenceanduniqueness, aswellaSthedecayestimateoftheenergy.

    3ExistenceandUniquenessofLocalSolution

    Inthissection,wegivethemaintheorembelowthatstatestheexistenceanduniquenessot themildsolution.Thedriftcoefficientfisassumedtobenon

    LipschitzianwhichWaNproposed

    inf61.Hence,wecouldnotconcludetheglobalsolutiondirectly.Inanotherway,theproof iss!cllitintotwosteps.First,theloca1solutionisobtainedinacompletedmetricspacein thissection.Then,followingtheestimationontheenergyfunctioninnextsection,theglobal existenceisproved.

    Theorem3.1Letfsatisfythefollowinghypotheses.

    (i)ForanyX?,thereexistsnconstantC>0suchthat

    ,()lc(1+Ixlp--1)Ix(3.1)

(ii)Foranyz,Y?,

    (iii)Forany?,

    ,(z)f(y)lco+jxtp+11)IxY

    T.WeiandM.Jiang

    '(z).,(p+1)F(z).(),F():=,()d

    (3.2)

    (3.3)

    /orsDmepositiveconstantP1,and(d2)pd.Assumeu0?D(),Ul?L2(D), (?):esuchthat.()dt<1.Thenthereexists0uniquemildsolutionutoequation

    (1.2)whichbelongsto([0,×Q,D(fA)).

    ProofWefirstsolvethetruncatedequationwithfsatisfyingtheglobalLipschitzcondition

    Forevery1,define?:0,.o)__?[0,1]aSaCfunctionsuchthat ?c=ifx

    >

    <

    n

    n'

    +

    (3.4)

    satisfiesthatIIInl1andl?,nI2.Wewillprovetheexistenceanduniquenessofthesolution

    tothefollowingstochasticintegralequation )=()uo+2aJot(r)u0dr+if0s(r)uldr-2a~0ts(r)"(r)dr

    j

    f0

    t

    1"S(tr)n(11(r)l1),((r))dr+fo*l,sr)dw(r).(3.5)

    )=()o,

    u

    )=@)uo+2a~ots(r)u0dr+1ts)uldr--2a~ots(r)

    Denote

,

    

    /1s(tI,

    0

    r)n(1lu(r)()l1),((r)())

    A:={"?(0,×Q;D())sup0<t<T

    +/1

    0

    )(r)

    (3.6)

    s(tr)dW(r).(3.7)

    EIIv~"(t)ll<?)

    equippedwiththedistancegeneratedby([0,T]×【2;D(v/-A)),i.e

    d(vl,2):supEIIv~vl(s)~/v2(s)

    0<——

    s

    <

    T

    Then(A,d)isacompletemetricspace.Wewillshowthat{u)isaCauchysequenceinAfor

    eachnandthelimitexistsandisasolutionto(3.5).Notetat4一?isnonnegativedefinite

    andunif0rm1yelliptic.BytheSob01evinequalityjthereenx.lsI~sa'a'k'c|6tant

    >0suchthat

    foranyV?D().FromTheorem2.1, ll

    Elive()))lI<ll+2E~ll+tE+ Cn".lI+IlUlll+?九)]<?i=0

    (3.8)

    Accordingto(2.3)inTheorem2.1'wehaveforanyV1,u2?Athat

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