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up of Solution for the Nonlinear Sobolev-Galpern Equation

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up of Solution for the Nonlinear Sobolev-Galpern Equationup,of,for,the,The

    up of Solution for the Nonlinear

    Sobolev-Galpern Equation

    Chin.Quart.3.o{Math

    2009,24(3):432438

    BlowupofSolutionfortheNonlinear

    Sobolev--GalpernEquation

    ZHANGEnbiao,JIANGChengshun

    (tit"ofElectroicTechnology,UniversityofInformnti.nEngineering,Zheng~h."450004,Chinn)

    Abstract:Inthispaper,theinitialboundaryvalueproblemsofthen0nlinearSob0lev- Galpernequationarestudied.Theexistence,uniquenessoflocalsolutionfortheproblem areobtainedbymeansofaspecialGreen'sfunctionandthecontractionmappingprinciple. Finally,theblow-upofsolutioninfinitetimeundersomeassumedc0nditionsisDrovedwith theaidofJensen'sinequality.

    Keywords:nonlinearSobolevGalpernequations;Green'sfunction;bl0w.

    up;Jensen'sin.

    equality

    2000MRSubjectClassification:35B40.35K57

    CLCnumber:0175.26Documentcode:A

    ArticleID:10020462(2009)03043207

    Thenonlinearequation

    ?1.Introduction

    vt(x,t)=o(v(z,t))+.?(,t)

    wasfirstlyderivedandstudiedbyPLDavis[in1972,whichwascalltheSobolev.GalperntvDe equation-Itisanexampleofageneralclassofpseudo-parabolicpartialdifferentialequations Showalter[2..

    Inmechanicsandphysics,ithasmanyimportantapplications.VlariousOfthe

    mixedboundaryvalueproblemscarlbeusedasthemathematicalmodelstodescribeafew physicalphenomena,suchas,modelsforwaterwavemovementsinhydrodynamicsTing[3]: modelsforseepageofhomogeneousfluidsthroughafissuredrockBarenblatt[4];modelsforthe

    heatconductioninvolvingtwotemperaturesinthermodynamicsChen[51. Theexistenceofuniquesmoothsolutionandtheconvergenceofthesolutionoftheinitial boundaryvalueproblemoftheequation(1.1)havebeenconsideredin1.Recently,thepaper

    Receiveddate:200612.01

    Biography:ZHENGEn'biao(1980

    ),male,nativeofFengqiu,Henan,alecturerofInformationEngineering M.S.D.,engagesininformationsecurity.

    NO.3ZHANGEn.-biaoetal:Blow?-upofSolutionfortheNonlinear

    Liu[extendedthoseresults.

    Inndimensionalcase.theexistenceanduniquenessoftheglobal

    generalizedsolutionhavebeenobtained.Thenon

    negativityofthesolutioncorrespondingto

    thenonnegativeinitialvalue,theasymptoticbehaviorandtheblow

    upofthesolutionhave

    alSObeendiscussedinf61.Forthecaseofonedimension,thepaperprovedtheexistenceand uniquenessofglobalstrongsolutionaslongaLs(s)isboundedfrombelow.Furthermore, documentShang[showedthenonexistenceofglobalsolutionsfortheinitialboundaryvalue problemsofthenonlinearSobolevGalpernequationwhen(s)isnotboundedfrombelow.

    Thepaper7]hasalsoconsideredtheinitialboundaryvalueproblemsoftheequation "IStQ""?:,(),

    andobtainedtheblowupofthesolutionsofthisproblems.

    Inthispaper,wearegoingtoinvestigatetheinitialboundaryvalueproblemofthegener

    alizedSobolevGalpernequation:

    tt=,(t),

    (,0)=uo(x),0z1,

    (0,t)=u(1,t)=0,t>0,

    wheref(u)isthegivennonlinearfunction,Uo(X)isthegiveninitialvaluefunction (1.2)

    (1.3)

    (1.4)

    Differentfromalltheabovedocuments.ouraimiStotakeadvantageoftheGreenfunctionto study(?).Firstly,theoriginalproblemistransformedintoanequivalentoperatorequationby Green'Sfunctionofaboundaryvalueproblemforasecondorderordinarydifferentialequation.

    Secondly,theexistenceanduniquenessofthegeneralized1ocalsolutioniSobtainedusingthe fixedpointtheoryofcontractingmappingsinaBanaehspace,underweakerhypothesestothe function,(u).Finally,theblowupofthesolutioniSdiscussedinthefinalsection.

    ?2.ExistenceandUniquenessofGeneralized

    LocalSolution

    Inthefollowingwewillreducetheproblem(1.2)(1.4)toanequivalentintegralequation

    byGreen'Sfunction.

    wefidenoteII"(~)llct.,

    1=0m<axIu(tobethenormofuinc[o,1].LetG(x,?)bethe

    Green'Sfunction(see

    8)oftheboundaryvalueproblemfortheordinarydifferentialequation a(x,)=

    v(x)3"()=,,v(0):y(1)=0,

    0<?,

    (2.5)

    z1.

    一一

    ?Z

    ee

    一一

    q

    ,,,

?

    一一

    ee

    一一

    ??

    ee

    一一

    一一

    ee

    一一

    ee

    11

    ,IIl,l??

    CHINESEQUARTERLYJOURNALOFMATHEMATICSVb1.24 ThentheGreen'Sfunctiona(x,)satisfiesthefollowingproperties

    Lem_ana2.1e(x,)satisfies:

    (i)a(x,?)iscontinuousinQ={01,01)

    (ii)v(z,):G(?,)

    (iii)0a(x,)<;,0z1,0?1.

    ProofObviously,theproperties(i)and(ii)hold.Weonlyprovetheproperty(iii).In

    fact,wefirstconsidera(x,?)on0.ItiseasytoknowthatG(x,)0.Since

    (e.e)<(efe'),

    a(x,)<e--e-1)(ee21_e-2~+l+e).

    Fromtheinequality(e2{+e1-2)2andequation(?)itfollowsthat a(x,f)<e2ee1

    2(ee)2(e+1)

    Usingtheproperty(ii)weseethatV0z (5)holds.

    2

    <'

(2.6)

    1,0?lithas0a(x,)<;.Theproperty

    SupposetI(z,t)isaclassicalsolutionoftheproblem(1.2)(1.4).Theequation(1.2)and theboundaryvaluecondition(1.4)Canberewrittenasfollows: ut(x,t)+f(u(x,t)),(0)】一【ut(x,t)+f(u(X,)),(0)zz=f(u(X,))1(o)(2.7)

    :

    t(z,t)+fCu(.,?)),(0)=/G(z,?)((?,t)),(0)d.

    Integratingtheaboveformulawithrespecttotwehave (,)=).

    ,

    ))+foxGtUo(Xf(u(xf(O)]dTa(x000,),(u(,)),(.)]d?d.(,)=)/,))+,)

    ,(u(,7-)),(0)]d?d.',

    (2.8)

    (2.9)

    Hence,thesolutionoftheproblem(1.2)(1.4)isthesolutionoftheintegralequation(2.9). Nowwearegoingtoprovetheexistenceanduniquenessofthelocalcontinuoussolutionfor

    integralequation(2.9)bythecontractionmappingprinciple. Wedenotethefunctionspace

    B(M,)={I(z,t)?(0,,C[0,1),(o,t)=(1,t)=0,IIII.,M+1),

    N0.3ZHANGEn.biaoetal:BlowupofSolutionfortheNonlinear???435 where10.=supsupIu(x,t)l,andMIlu0(z)llc[o,1].ItiseasytoseethatB(M,T)is

    O<?<T0<x<1

    anonemptyBanachspaceequippedwiththenorm0.TforeachM,T>0.

    WedefinethemapSasfollows:

    ,

    ):)/0If(u(z,))+/olGSw(xtUo(Xf(0)]dTa(x,f)[,((f,.r))f(0)]d~dT.(2.10J00),):)

    u(z,))+,f)[,((f,.r)).(2.)

    OurgoalistoshowthatShasauniquefixedpointinthespaceB(M,). Lemma2.2Assumethat()?el0,1],uo(0)=u0(1)=0,/(8)?C,thenSmaps

    B(M,T)intoB(M,T)andthemapS:B(M,T)卜?B(M,T)isstrictlycontractiveifTis appropriatelysmallrelativetoM.

    ProofLet,()=m{l,(s)l,lf(s)I),Vr]0.From(2.10)andLemma2.1wecangetI"

    IIS03110.TM+7T-](M+1).

    IfTsatisfies

    ,(2.11)

    thenl130311~,TM+1.Therefor,if(2.11)holds,thenSmapsB(M,T)intoB(M,). NextwearegoingtoprovethatthemapSisstrictlycontractive.LetT>0andV031,032?

    B(M,),wehave

    1--$032-~---/o'If((,)),((,))d

    +

    tr1

    G(z,(()f(03z()?

    UsingthemeanvaluetheoremandCauchy'Sinequality,weobtain 1[$031-l10,T;(+1)I1031-~ll.

    ?Tsatisfies

    n{l_,

    thenlIslS~21lo,TPIIu12I10.T,whereP

    ,

    )'

    0.Thelemmaisproved.

    ItfollowsfromLemma2.2andthecontractionmappingprinciplethatforappropriately

    chosenT>0,Shavefixedpoint(z,t)?B(M,),whichisageneralizedsolutionofthe problem(1.2)(1.4).Sowecansummarizethisresultasfollows: Theorem2.1AssumethattheconditionsofLemma2.2hold.Thentheproblem (1.2)(1.4)hasauniquegeneralizedlocalsolutiont'(,t)?([0,To),el0,1),where0,To)

    isamaximaltimeinterva1.

    ThemaximumvalueofToinTheorem2.1iscalledthelifespanofthesolution,ingenera1.

    ThefollowingtheoremtellsUSthatthesolutionCanextendedtothewholetimeintervalwhen

    thesolutionsatisfiessomeconditions.

    CHINESEQUARTERLYJOURNALOFMATHEMATICSV01.24 Theorem2.2

    To00

    AssumetheTheorem2.1holds,andifsupIIu(?,t)llco'1<co,then

    te[o,To】一

    ProofSupposethatu(x,t)?(0,To),C[0,1)isthelocalsolutionfortheproblem (1.2)(1.4),0,To)isthemaximaltimeintervalofexistencefort(,t)?B(M,),andTo<?.

    ForanyT?【0,To),weconsidertheintegralequation: ,)="(z,)一【,))+

    c

    I

    Gv(xtf(v(xf(0)]dTG(x,)[,(,))f(0)]d~dT.(2.13

    ooJo

    ),)="(z,)/,7-))+/,)[,(,7-)).(2.).,J

    ByvirtueofsupIlu(?,t)llc0'1<.o,wecanchooseappropriatelyT?【0,To)suchthatfor eachT?【0,To)theintegralequation(2.13)hasauniquesolutionv(x,t)?([0,T}),el0,1]) usingthesamemethodabove.Inparticular,(2.12)revealsthatTcanbeselectedindepen-

    dentlyofT?【0,To).

    SetT,_To,letdenotethecorrespondingsolutionof(2.13),anddefine

    ,t)=t?【0,T;

    t?IT,+孚】.(2.14)

    Obviously,isasolutionoftheequation(2.13).Bylocaluniqueness,ityields:u.This violatesthemaximalityto0,To).Hence,ifsupIlu(?,t)llc0,l<coholds,thenTo=?.This te[o,To】一

    completestheproofofthetheorem.

    ?3.Blow-up

    Inthissectionwearegoingtoconsidertheblowupofthesolutionfortheproblem

    (1.2)(1.4).Wefirstgivethefollowinglemma.

    Lemma3.1(JenseninequalityBellman[.)If9()isacontinuous,realvaluedfunc tionontheintervala,6,/(x)isaconvexfunctiononR,q(x)?L[0,6,andq(x)0,q()?0, then

    rb1b

    Nowwehave

    Theorem3.1

    9()g(z)d

    b

    q()d

    f(g(x))q(x)dx

    q(x)dx

    Letu(z,t)betheclassicalsolutionoftheproblem(1.2)(1.4).Suppose thatthefollowingconditionsaresatisfied: (i)=一号uo()sinrxdx>0,wheresin~rxdenotesthefirstnormalizedeigenfunction

    fortheproblem

    u"+=0,u(0)=u(1)=0,0<z<1

    ,J

    ,.:

    t?

    ,,

    ,L,LU

    ,?lJ'l??\

    6/o

    NO.3ZHANGEn_bia0etal:BlowupofSolutionfortheNonlinear437 and=7r2denotesthecorrespondingfirsteigenvalue;

    (ii),(s)isalocalLipschitzcontinuousfunction,,(s)2

    satisfying(s)>0,Vs?[yo,+?),o.<+?,where

    (iii)Vz?(0,1),t10(z)>0andt'0(z)?0?

    9(s).

    Yo

g(s)isaconvexfunctionand

    

    uo(x)sinrxdx;

    Thenthesolutionoftheproblem(1.2)(1.4)blowupinfinitetime,i?e?,thereexista

    constantT<?,suchthatllullL2..whentT?

    ProofLet(t):一号foU(X,t)sinzd?.Multiplytheequati.n(1-2)bysinzd

    inteateorer(0,1).Integratingbypartswithrespecttoweobt

    (1+zr2)17=/0,(u)sinzd.

    nomJensen'Sinequalityandthecondition(ii),wehave (1+7r2)/0g()sin~rxdx?不29()(3.15)

    Since:一号u.(z)sinrcxdx>0,bytheconditi.ns(i)?(iii)and(3?5),>0wehave (t)>Yo>0andO(t)>0.Soweget

    >dt.(3.16)0,,

    Integratetheinequality(3.16)over(0,T)weobtain asI+7~2d8<

    Finally1since17(t)>0,whent_?,wehave(t)_?+.o,whichimpliesthatllt2_?+..?

    Thetheoremiscompleted.

    Usingthesamemethod,wealsohave

    Theorem3.2Assumetheequation(1.2)satisfiestheinitialvaluecondition(1?3)and

    thefollowingsecondboundaryvaluecondition: u(O,t)=u(1,t)=0,t>0.(3.17)

    Let(,t)betheclassicalsolutionoftheinitialandboundar'Yvalueproblem(1?2),(1?3),

    (3.17).Supp0sethat,(s)isalocalLipschizcontinuousfunction,,(8)?9(s)_9(s)isacon

    functionandsatisfying9(s)>0,Vsso,Joo<+?,wherey0=一号u0(z)cosrcxdx> 0;v?(0,1),o(z)>0and0()?0.Thenthesolutionoftheproblem(1.2),(1?3),(3?17)

    blowupinfinitetime,i.e.,thereexistaconstantT>oo,suchthatlI~JiL2_?+.owhen

    ProofLet

    integrateover(0,

    tozweobtain

(1+7r2)=,()c.szd.

    1

    p

    .m

    ?

    .

    ?

    -2i

    :

    438CHINESEQUARTERLYJOURNALOFMATHEMATICS,,01.24 FromJensen'Sinequalityandtheconditionoftheorem, wehave

    (1+.)17/o9()cos7rxdxr2g().

    Sincey0=一三0()cosrcxdx>0,Vt>0wehave(t)>>0and(t)>0.Nowintegrate

    theinequality(1+7l-2)71-2g(y)over(0,T),weobtain

    7rz

    .

    ,…而d8=<+?.

    Thisimpliesthatthemaximaltimeinterval(0,T)ofexistencefory(t)isfinite.Finally,

    sincey(t)>0,wehavey(t)-?+?(t.?T),i.e.,

    whichprovesthetheorem.

    2

    -?+?,

    References]

    1DAVISPL.Aquasilinearparabolicandarelatedthirdorderproblem[J].NJMathAnalAppl,1

    972,40

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