USF Libraries
USF Digital Collections

Longtime dynamics of hyperbolic evolutionary equations in ubounded domains and lattice systems

MISSING IMAGE

Material Information

Title:
Longtime dynamics of hyperbolic evolutionary equations in ubounded domains and lattice systems
Physical Description:
Book
Language:
English
Creator:
Fall, Djiby
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
Global attractor
Wave equation
Absorbing set
Asymptotic compactness
Lattice system
Dissertations, Academic -- Mathematics -- Doctoral -- USF   ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: This dissertation is a contribution to the the study of the longtime dynamics of evolutionary equations in unbounded domains. It is of particular interest to prove the existence of global attractors for solutions of such equations. Th this end one need in general some type of asymtotical compactness. In the case the evolutionary PDE is defined on a bounded domain, asymptotical compactness follows from the regularity estimates and the compactnes of Sobolev embeddings and therefore the existence of attractors has been established for most of the disipative equations of mathematocal physics in a bounded domain. The problem is more challenging when the domain is unbounded since the Sobolev embeddings are no longer comapct, so that the usual regularity estimates may not be sufficient.To overcome this obstacle of compactness, A.V. Babin and M.I. Vishik introduced some weighted Sobolev spaces. In their pioneering paper, Proc. Roy. Soc. Edinb.
Thesis:
Thesis (Ph.D.)--University of South Florida, 2005.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Djiby Fall.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 67 pages.
General Note:
Includes vita.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001681065
oclc - 62735248
usfldc doi - E14-SFE0001053
usfldc handle - e14.1053
System ID:
SFS0025374:00001


This item is only available as the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001681065
003 fts
005 20060215071205.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 051222s2005 flu sbm s000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0001053
035
(OCoLC)62735248
SFE0001053
040
FHM
c FHM
049
FHMM
090
QA36 (Online)
1 100
Fall, Djiby.
0 245
Longtime dynamics of hyperbolic evolutionary equations in ubounded domains and lattice systems
h [electronic resource] /
by Djiby Fall.
260
[Tampa, Fla.] :
b University of South Florida,
2005.
502
Thesis (Ph.D.)--University of South Florida, 2005.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 67 pages.
Includes vita.
3 520
ABSTRACT: This dissertation is a contribution to the the study of the longtime dynamics of evolutionary equations in unbounded domains. It is of particular interest to prove the existence of global attractors for solutions of such equations. Th this end one need in general some type of asymtotical compactness. In the case the evolutionary PDE is defined on a bounded domain, asymptotical compactness follows from the regularity estimates and the compactnes of Sobolev embeddings and therefore the existence of attractors has been established for most of the disipative equations of mathematocal physics in a bounded domain. The problem is more challenging when the domain is unbounded since the Sobolev embeddings are no longer comapct, so that the usual regularity estimates may not be sufficient.To overcome this obstacle of compactness, A.V. Babin and M.I. Vishik introduced some weighted Sobolev spaces. In their pioneering paper, Proc. Roy. Soc. Edinb.
590
Adviser: Yuncheng You, Ph.D.
653
Global attractor.
Wave equation.
Absorbing set.
Asymptotic compactness.
Lattice system.
690
Dissertations, Academic
z USF
x Mathematics
Doctoral.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.1053



PAGE 1

LongtimeDynamicsofHyperbolicEvolutionaryEquationsinUnboundedDomainsandLatticeSystemsbyDjibyFallAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:YunchengYou,Ph.DAthanassiosKartsatos,Ph.DWen-XiuMa,Ph.DMarcusMcWaters,Ph.DDateofApproval:April7,2005Keywords:globalattractor,waveequation,absorbingsetasymptoticcompactness,latticesystemcCopyright2005,DjibyFall

PAGE 2

Speciallydedicatedtobothmyparentswhohavealwaysbeenasourceofinspirationandencouragementsincerstgrade.DiaradieufYaaybooy.

PAGE 3

ACKNOWLEDGEMENTSIexpressmyprofoundgratitudetomysupervisorProfessorYunchengYouforpropos-ingmesuchaninterestingtopic.Hisguidancewithenthusiasticencouragementshasbeensupportiveduringthetimeofmypreparation.Morethenonce,Ihavebeeninspiredbyhisoriginalideasthroughoutthecourseofthisresearch.MywarmestappreciationsalsogotoProfessorsAthanassiosKartsatos,Wen-XiuMaandMarcusMcWatersforacceptingtobepartofthethesupervisingcommittee.Ihavegreatlybenetedfromtheirconstantconsiderationformyprogress.IwouldliketothankProfessorDavidRabsonofthePhysicsDepartmentforkindlyacceptingtobethechairpersonofthedefensecommittee.Hiscommentsandsuggestionsonmyworkhavebeenavaluablesourceofinspiration.IamdeeplyindebtedtothemathematicsdepartmentattheUniversityofSouthFloridaforitsgenerousnancial,mathematicalandpersonalsupportduringallthese4yearsofgraduatestudies.Thankstoallthosewho,bytheirfriendshiporbytheirencouragements,havecontributedtoanappropriateframeofmindforthiswork.Icannotndtherightwordstothankmylovingparentsandallmyfamilywhoprovided,astheyhavethroughoutmylife,akindconcernandcarethathasalwaysstimulatedandsustainedme.

PAGE 4

TableofContentsAbstractii1INTRODUCTIONANDGENERALCONCEPTS11.1SemiowsandAttractors.............................41.2EvolutionaryEquationsandSemigroupTheory.................61.2.1SemigroupsofLinearOperators......................61.2.2NonlinearEvolutionEquations......................101.3SomeUsefulInequalities..............................122ATTRACTORSFORDAMPEDWAVEEQUATIONS142.1TheWaveEquationwithMassTerm.......................152.1.1ExistenceofSolutionsandAbsorbingSet................152.1.2GlobalAttractor..............................212.2TheWaveEquationWithoutMassTerm.....................333DYNAMICSOFSECONDORDERLATTICESYSTEMS393.1TheExistenceandBoundednessofSolutions..................403.1.1TheExistenceandUniquenessofSolutions...............413.1.2TheBoundednessofSolutions.......................433.2GlobalAttractor..................................464FINALREMARKS52References57AbouttheAuthorEndPagei

PAGE 5

LongtimeDynamicsofHyperbolicEvolutionaryEquationsinUnboundedDomainsandLatticeSystemsDjibyFallABSTRACTThisdissertationisacontributiontothestudyoflongtimedynamicsofevolutionaryequationsinunboundeddomainsandoflatticesystems.Itisofparticularinteresttoprovetheexistenceofglobalattractorsforsolutionsofsuchequations.Tothisend,oneneedsingeneralsometypeofasymptoticalcompactness.InthecasethattheevolutionaryPDEisdenedonaboundeddomaininspace,asymptoticalcompactnessfollowsfromtheregularityestimatesandthecompactnessoftheSobolevembeddingsandthereforetheexistenceofattractorshasbeenestablishedformostofthedissipativeequationsofmathematicalphysicsinaboundeddomain.TheproblemismorechallengingwhenisunboundedsincetheSobolevembeddingsarenolongercompact,sothattheusualregularityestimatesmaynotbesucient.Toovercomethisobstacleofcompactness,A.V.BabinandM.I.VishikintroducedsomeweightedSobolevspaces.Intheirpioneeringpaper[2],theyestablishedtheexistenceofaglobalattractorforthereaction-diusionequationut)]TJ/F24 10.909 Tf 10.909 0 Td[(u+fu+u=g;x2RNLately,anewtechniqueof"tailestimation"hasbeenintroducedbyB.Wang[49]toprovetheexistenceofglobalattractorsforthereaction-diusionequationintheusualHilbertspaceL2RN.Inthisresearchwetakeonthesameapproachtoprovetheexistenceii

PAGE 6

ofattractingsetsforsomenonlinearwaveequationsandhyperboliclatticesystems.Thedissertationisorganizedasfollows.IntherstpartChapter2,weprovetheexistenceofaglobalattractorinH10RNL2RNforthewaveequationutt+ut)]TJ/F15 10.909 Tf 10.909 0 Td[(u+u+fu=g;t>0;x2RN:Removingthecoercivemasstermufrom,weachievethesameresultforthemorechallengingequationutt+ut)]TJ/F15 10.909 Tf 10.909 0 Td[(u+fu=g;t>0;x2whereisadomainofRNboundedonlyinonedirection.Thesecondpartofthedissertationdealswithsomelatticesystems.WeestablishinChapter3theexistenceofglobalattractorfortheequationui+_ui)]TJ/F15 10.909 Tf 10.909 0 Td[(ui)]TJ/F22 7.97 Tf 6.586 0 Td[(1)]TJ/F15 10.909 Tf 10.909 0 Td[(2ui+ui+1+fui=gi;i2Zwhichisaspatialdiscretizationof.iii

PAGE 7

1INTRODUCTIONANDGENERALCONCEPTSHenriPoincare-1912isoftenreferredtoasthefatherofnonlineardynamics".To-wardtheendofthenineteenthcentury,heforthersttimepointedoutthatirregularbehaviorsinmechanicsarenotatallanunusualfeatureifthesystembeingstudiedinvolvesanonlinearinteraction.Verysimplesystemscanhavehighlycomplexdynamics.Thebe-haviorofthesolutionsofsuchsystemsoverlongertimeisquiteirregularandpracticallycannotbepredicted.SincePoincarethestudyofnonlineardynamicsstartedtodevelop,withmajorcontributionsofgreatmathematiciansandphysicistssuchasLyapunov-1918andBirkho1884-1944whodevelopedtheconceptofdynamicalsystemsasweknowittoday.Meanwhilethestudyofnonlineardynamicsextendsfarbeyondmechanicstomanyeldsnotonlyofphysicsbutalsoofchemistry,biology,economicsetc.Before1950,itprimarilyfocusedonthenitedimensionalsystemsusuallymodeledbyordinarydierentialequations.Thetheoryfortheevolutionarypartialdierentialequationswasslowertoemerge.Thistheoryalongwiththestudyofdierential-delayandlatticesystemsconstitutewhatisknownasinnitedimensionaldynamicalsystems.Theliteratureisextensiveonthestudyofthesesystems,mainlyforpartialdierentialequationsinaboundeddomain,seeforin-stanceJ.Hale[23],G.Sell&Y.You[42]orR.Temam[47],andthereferencestherein.Itisjustrecently,sincethepioneeringworkofBabinandVishik[2],thatmathematiciansgotinterestedinthedynamicsofpartialdierentialequationsinunboundeddomains.Inthisdirection,severaltypesofevolutionaryequationshavebeeninvestigatedwithinter-estingresults;yettherearestilllotsofopenproblemsandlargeroomformathematicalcontributions.Onecanbeinterestedindierentfeaturesinthestudyofthedynamicsofasystem:1

PAGE 8

singularityformations,nite-timeblowup,existenceofchaos,attractingsets,bifurcationtheory,invariantmanifolds,exponentialdichotomies,stability,...etc.Inthisworkwefocusonthelongtimebehaviorofthesolutionsofevolutionaryequationsinunboundeddomainsandoflatticesystems.Inparticular,thetopicistheexistenceofglobalattractors.Theglobalattractorisacompactinvariantsetattractingthetrajectorybundlesofallboundedsubsetsastimegoestoinnity.Therefore,ifitexists,aglobalattractorcontainsalltheessential,permanentdynamicsofthesystem.AndveryoftentheglobalattractorhasnitefractalorHausdordimension,thusreducingtheinitialinnitedimensionalproblemtoanitedimensionaloneinthelongrun.Inthecurrenttheoryofinnitedimensionaldynamicalsystems,theglobalattractorisahighlighted,coretopic.Theexistenceofglobalattractorsfordissipativesystemsfollowsingeneralfromsometypeofasymptoticalcompactnessofthecorrespondingsemiow.ThisisprovedincasethedomainisboundedbyaprioriestimatesandthecompactnessofSobolevembedings.ThismethodseemsnottoworkwhenthedomainisunboundedsincetheSobolevembeddingsarenolongercompact.Itthenbecomesadiculttasktodealwiththiscompactnessissue.Twomajortechniquesseemtoworkinovercomingthisdiculty:workingwithweightedSobolevspacesasphasespaceorusingthetailestimationmethods".In1990BabinandVishik[2]forthersttimeshowedtheexistenceofaglobalattractorforthereaction-diusionequationinRN,ut)]TJ/F24 10.909 Tf 10.909 0 Td[(u+fu+u=g:.1TothisendtheyusedtheweightedSobolevspaceL2RNasphasespace.Theyproved,for<0,theexistenceofaglobalattractorintheweaktopologyundercertaingrowthconditionsonthenonlinearityf.For>0,theattractorisinthestrongtopologyandifgxdecreasessucientlyfast,thentheattractorsfordierentdonotdependon.In1994Feireisl,Laurencot,SimondonandToure[20]consideredasimilarequationas.1withoutthemasstermuandtheyshowedthatfor<)]TJ/F25 7.97 Tf 9.681 4.295 Td[(N 2,theattractorisindeedinthestrongtopologyofL2RN.Someotherauthorshavealsoconsideredweightedspacesfordierenttypesofequations.2

PAGE 9

Forthewaveequationutt+ut)]TJ/F24 10.909 Tf 10.91 0 Td[(xu+fu=x;x2RN;t>0;N.I.KarachaliosamdN.M.Stavrakakis[24]establishedtheexistenceandnitedimension-alityofaglobalattractorinL2gRNwheregx=1 x.Workingwiththeweightedspaceshasadisadvantageofrestrainingthechoiceoftheinitialdata.In1999,B.Wang[49]cameupwiththenewideaoftailestimations"toprovetheasymptoticcompactnessofthesemiowgeneratedbythereaction-diusionequation.1.ThisledtotheexistenceofaglobalattractorinL2RNforsystem1.1.ThismethodfeaturesanapproximationofRNbysucientlylargeboundeddomainskandthenshowthenullconvergenceofthesolutionsinRn)]TJ/F15 10.909 Tf 11.147 0 Td[(k.Thismethodhasbeenusefultostudyingthedynamicsofmanyevolutionaryequationsinunboundeddomainsaswellaslatticedynamicalsystemsseeforinstance[4,49,43,50,28].ThedynamicsofnonlinearwaveequationsinunboundeddomainshavebeenextensivelystudiedbyEduardFeireisl.In[16]heshowedtheexistenceofglobalattractorfor3DwaveequationinH1R3L2R3,butforthen-dimensionalproblemtheattractorisonlylocallycompact,see[17].Similarresultshavebeenobtainedin[53].ItseemslittleisknownontheexistenceofglobalattractorinthetraditionalSobolevspacesfornonlinearwaveequationsinunboundeddomains.InChapter2,weconsiderthedampedwaveequationwithmassterm,utt+ut)-222(4u+u+fu=gx;x2RN;t>0.2andshowtheexistenceofglobalattractorforthecorrespondingdynamicalsystem.Thisisdonebyapplyingthegeneralizedtailendestimation"methodintroducedin[4]and[49].Moreover,removingthemasstermin1.2,weachievetheexistenceresultfortheequationutt+ut)-222(4u+fu=gx;x2;t>0.3whereisadomainofRN,boundedonlyinonedirection.ThisfeaturestheuseofthePoincareinequalitytoachievethemonotonicityofthelinearoperatorforthecorresponding3

PAGE 10

transformedrst-orderproblem.InChapter3westudythedynamicsoftheseconderorderlatticesystem,withoutmassterm,ui+_ui)]TJ/F15 10.909 Tf 10.909 0 Td[(ui)]TJ/F22 7.97 Tf 6.587 0 Td[(1)]TJ/F15 10.909 Tf 10.909 0 Td[(2ui+ui+1+fui=gi;i2Z.4whichcanbeseenasaspatialdiscretizationof.3inonedimension.Inchapter4wegivesomeremarksonthedimensionftheglobalattractorsandpresentsomenewdirectionswithinterestingopenproblems.Beforewegetintothedetailsofourwork,letusintroducerstthebasicdenitionsandresultsrelevanttothegeneraltheoryofdynamicalsystems.WepresentinthisintroductorychapterthenotionsofsemiowsandattractorsalongwithabriefpresentationofthetheoryofsemigroupsanditsrelationtosolvingabstractnonlinearequationsinaBanachspace.Wealsogiveanexampleofasectorialoperatorandwenishwithsomeinequalitiesthatwillbeusefulinthesubsequentchapters.1.1SemiowsandAttractorsWewilluseinthisworkthedenitionofsemiowsasinTemam[47].AstrongerversioncanbefoundinSell&You[42],wheretheonlydierenceisthecontinuityproperty.Denition1.1.1LetH;dbeacompletemetricspace.AfamilyofoperatorsfStgt0iscalledasemiowonH,ifitsatisesthefollowingproperties:1.S=IidentityinH,i.e.Su=u8u2H,2.StSs=St+s;8s;t2R+,3.ThemappingSt:H!Hiscontinuousforeveryt0.WeintroducenowtheconceptsofinvariantsetsandattractorsofasemiowDenition1.1.2LetStbeasemiowonHandKH.WesaythatKispositivelyinvariantifStKK,forallt0.KisinvariantifStK=K,forallt0.4

PAGE 11

TodeneattractorswewillneedthefollowingasymmetricHausdorpseudodistance:supa2Ainfb2BdA;b.5whereA;BareboundedsetsinH.WesaythatAattractsBifhStB;A!0;ast!1;.6thatis:forevery">0,thereexistsT0suchthatdStu;A",foralltTandu2B.Denition1.1.3AsubsetAofHiscalledanattractorforthesemiowStprovidedthat1.Aisacompact,invariantsetinH,and2.thereisaneighborhoodUofAinHsuchthatAattractseveryboundedsetinU.AnattractorAthatattractseveryboundedsetinHiscalledaglobalattractor.Theexistenceofglobalattractorisingeneralrelatedtowhatsomeauthorscallthedissipativity"ofthedynamicalsystem.Thisisequivalenttotheexistenceofabsorbingsets.Denition1.1.4LetBbeasubsetofHandUanopensetcontainingB.WesaythatBisanabsorbingsetinUiftheorbitofanyboundedsetinUentersintoBafteranitetimewhichmaydependontheset:8<:8B0U;B0bounded9t1B0suchthatStB0B;8tt1B0:WealsosaythatBattractstheboundedsetsofU.Wehavealsotherelatedconceptofasymptoticalcompactness.Denition1.1.5AsemiowfStgt0issaidtobeasymptoticallycompactonUifforeveryboundedsequencefunginUandtn!1,fStnungt0isprecompactinH.5

PAGE 12

Wearenowreadytopresentastandardresultontheexistenceofglobalattractorswhichcanbefoundin[23,47].Theorem1.1.1LetfStgt0beasemiowinX.IffStgt0hasaboundedabsorbingsetandisasymptoticallycompactinH,thenfStgt0possessesaglobalattractorwhichisacompactinvariantsetthatattractseveryboundedsetinH.1.2EvolutionaryEquationsandSemigroupTheoryInpracticesemiowsaregeneratedbythesolutionsofdierentialequations.Wewillcon-siderabstractnonlinearODEsoftheformdu dt+Au=Fu;t.7inaBanachspaceX,whereAisanunboundedlinearoperatorinXandF:XR!Xisanonlinearfunctional.Inthissectionwewillpresentthegeneralexistencetheoryforequationssuchas.7.Thiswillapplydirectlytoawiderangeofevolutionarypartialdierentialequations.Wewillrstgivesomebasicnotionsonsemigrouptheorywhichisrelatedtosolvingthecorrespondinglinearproblemdu dt+Au=0:.81.2.1SemigroupsofLinearOperatorsIntheremainderofthissection,XdenotesaBanachspacewithnormkkXandLXisthespaceofboundedlinearoperatorsonX.Denition1.2.1WewillsaythatafamilyofoperatorsfTtgt0isaC0-semigroupoflinearoperatorsonX,ifTt2LXforallt2[0;+1andthefollowinghold:iT=IidentityinXiiTtTs=Tt+s;s;t2[0;+1iiilimt!0+Ttx=x,forallx2X.6

PAGE 13

WeseethataC0-semigroupisatypicalexampleofasemiowonX.Denition1.2.2LetTtbeaC0-semigrouponX,itsinnitesimalgeneratoristhelinearoperatorAonXdenedasfollowsThedomainofAis:DA=fx2X:limh!0+Th)]TJ/F24 10.909 Tf 10.909 0 Td[(I hxexistsinXgforx2DAweset:Ax=limh!0+Th)]TJ/F24 10.909 Tf 10.909 0 Td[(I hx=d+Ttx dtjt=0:NextwewillgiveanecessaryandsucientconditionforanoperatortobetheinnitesimalgeneratorofaC0-semigroupinaHilbertspaceH.Weneedtointroducerstsomeconcepts.LetHbeaHilbertspacewithinnerproducth;i.AlinearoperatorA:DAH!HissaidtobeaccretiveifRehAx;xi0;8x2DA:IfinadditionwehaveRI+A=HrangeofI+AisequaltoHthenwesaythatAismaximalaccretive.AC0-semigroupissaidtobenonexpansiveifkTtk1foreveryt0.Theorem1.2.1Lumer-PhillipsLetHbeaHilbertspace.Thenalinearoperator)]TJ/F24 10.909 Tf 8.484 0 Td[(A:DAH!HistheinnitesimalgeneratorofanonexpansiveC0-semigroupe)]TJ/F25 7.97 Tf 6.587 0 Td[(AtonHifandonlyifboththefollowingcondtionsaresatised:AisaclosedlinearoperatorandDAisdenseinH,andAisamaximalaccretiveoperator.Thisisaclassicalresultonsemigroupsandtheirgenerators.Theproofcanbefoundin[38,42].HoweverthisresultappliesonlytoHilbertspaces;thereisamoregeneraloneonBanachspaces,namelytheHille-Yosidatheorem.7

PAGE 14

Aswementionedinthebeginningofthissection,thesemigrouptheorywillallowustosolvethelinearproblem1.8.IndeedletAbethegeneratorofaC0-semigroupTtonX;itisshownthatforeveryx02DA,Ttx0isasolutionofequation.8intheclassicalsense.Forx02Xwecallxt=Ttx0amildsolutionof.8.TheexistenceofsolutionsintheclassicalsenseisrelatedtothedierentiabilityoftheC0-semigroupandthereisaparticularclassofdierentiablesemigroupscalledanalyticsemigroups.Denition1.2.3WewillsaythatTtisananalyticsemigroupinXifthereisanextensionofittoamappingTzdenedonsomesector[f0gsuchthat:Tz1+z2=Tz1Tz2forallz1andz2in[f0g,foreachx2X,onehasTzx!xasz!0in[f0g,foreachx2X,thefunctionz!TzxisananalyticmappingfromintoXwherethesectorisdenedas=fz2C:jargzj<;z6=0g;for2;:Arelatedconceptisthatofsectorialoperators.Denition1.2.4AlinearoperatorA:DAX!XissaidtobeasectorialoperatoronXifitsatisesthefollowing:Aisdenselydenedandclosed,thereexistrealnumbersa2R;2; 2andM1suchthatonehasaAandkR;AkM j)]TJ/F24 10.909 Tf 10.909 0 Td[(aj;forall2a.9whereAistheresolventsetofA,R;Atheresolventoperatorandadenedas:a=fz2C:jargz)]TJ/F24 10.909 Tf 10.909 0 Td[(aj>;z6=ag:Thenexttheoremwhichcanbefoundin[38,42]givestherelationbetweenanalyticsemi-groupsandsectorialoperators8

PAGE 15

Theorem1.2.2LetTtbeaC0-semigrouponXwithinnitesimalgeneratorAandletM1,a2RbechosensothatkTtkMe)]TJ/F25 7.97 Tf 6.587 0 Td[(at,forallt0.Thenthefollowingstatementsareequivalent:TtisananalyticsemigroupandthereisananalyticextensionsemigroupTzdenedonsomesector[f0gwith0<< 2,andaconstantM1MsuchthatkTzkM1e)]TJ/F25 7.97 Tf 6.586 0 Td[(aRezforz2.AisasectorialoperatorandonehaskR;AkM2 j)]TJ/F24 10.909 Tf 10.909 0 Td[(aj;forall2a;.10forappropriateconstantsM21and2; 2.Moreover,Ttisadierentiablesemigroup.Formanypartialdierentialequations,particularlytheparabolicones,thecorrespondinglinearoperatorissectorial.Inthefollwingexamplewepresentsomeellipticoperatorsthatturnouttobesectorial.Thiswillbeusedinthenextchaptertoshowthatthetransformedlinearoperatorforthenonlinearwaveequationismaximalaccretive.Example1.2.1LetbeeitherRnoranopenboundedsubsetofRnwithuniformlyC2boundary@.WeconsiderasecondorderdierentialoperatorAx;D=nXi;j=1aijxDij+nXi=1bixDi+cxIwithrealuniformlycontinuousandboundedcoecientsaij;bi;c.Weassumethatthematrix[aij]issymmetricandthatitsatisestheuniformellipticityconditionnXi;j=1aijijjj2;x2;2Rn;.11forsome>0.Moreoverif6=Rn,weconsiderarstorderdierentialoperatoractingontheboundaryBx;D=nXi=1ixDi+xI:.129

PAGE 16

Weassumethati;,belongtoUC1,andthattheuniformnontangentialityconditioninfx2@jnXi=1ixxj>0.13holds,withxbeingtheexteriorunitnormalvectorto@atx2@.LetX=Lp;1
PAGE 17

WeconsiderthefollowinginitialvalueproblemintheBanachspaceX:8<:du dt+Au=Fuut0=u02X;tt00:.14AssumethatthenonlinearityFbelongstoF2CLip=CLipX;X,thecollectionofallcontinuousfunctionsG:X!XthatareLipschitzcontinuousoneveryboundedsetBinX.Wesupposealsothat)]TJ/F24 10.909 Tf 8.485 0 Td[(AgeneratesaC0-semigroupTtonX.Atrst,wegivedierentnotionsofsolutionforproblem.14andthenpresentsomeexistenceresultsforsuchtypesofsolutions.Denition1.2.5LetI=[t0;t0+beanintervalinR+,where>0.Astronglycontinuousmappingu:I!Xissaidtobeamildsolutionof.14inXifitsolvesthefollowingintegralequationut=Tt)]TJ/F24 10.909 Tf 10.909 0 Td[(t0u0+Ztt0Tt)]TJ/F24 10.909 Tf 10.909 0 Td[(sFusds;t2I:.15IfuisdierentiablealmosteverywhereinIwithut;Au2L1locI;X,andsatisesthedierentialequationdu dt+Au=a:e:Fu;ont0;t0+;andut0=u0;.16thenuiscalledastrongsolutionof.14.Ifinaddition,onehasut2CI;Xandthedierentialequationin1.16issatisedfort00,theInitialValueProblem.14hasauniquemildsolutionuinXonsomeintervalI=[t0;t0+,forsome>0.AssumethatX=HisaHilbertspaceorareexiveBanachspace.Ifu02DAorTtisadierentiablesemigroup,thenthemildsolutionisastrongone.11

PAGE 18

Remark1.2.1ThesolutionuinTheorem1.2.3canbeextendedtoamaximumpossibleintervalI.Indeeduismaximallydeinedifeither=+1orlimt!)]TJ/F16 10.909 Tf 8.072 5.637 Td[(kutkX=+1.1.3SomeUsefulInequalitiesWepresentinthissectionsomeinequalitiesthatwillbeusedintheconsequentchapters.ThemostusedinequalitythroughoutourworkistheGronwallinequalitywhichcomesindierentforms.Wepresentheresomevariantsofit.Lemma1.3.1TheGronwallinequalitySupposethataandbarenonnegativecon-stantsandutanonnegativeintegrablefunction.Supposethatthefollowinginequalityholdsfor0tT:uta+bZt0usds:.17Thenfor0tT,wehaveutaebt:.18Lemma1.3.2TheuniformGronwallinequalityLetg;h;ybenonnegativefunctionsinL1loc[0;T;R,where0
PAGE 19

Remark1.3.1ThePoincareinequalityisusuallypresentedforboundeddomainsbuttheproofrequiresonlytheboundednessinonedirectionxi.13

PAGE 20

2ATTRACTORSFORDAMPEDWAVEEQUATIONSWestudyinthischaptertheexistenceofaglobalattractorforthefollowingtwodampednonlinearwaveequationsinanunboundeddomainofRN:utt+ut)-222(4u+u+fu=gx;t>0.1andutt+ut)-222(4u+fu=gx;t>02.2whereisapositiveconstant,gisagivenfunctionandfisanonlineartermsatisfyingsomegrowthconditionstobespeciedlater.Thelong-timebehaviorofsolutionsofsuchequationsinaboundeddomainwasstudiedbymanyauthors,forinstancein[47],[42]andthereferencestherein.Intheunboundeddomaincase,therealsoexistsanextensiveliterature.In1994,E.Feireisl[16]showedthatthemorechallengingequation.2admitsaglobalattrac-torinH1R3L2R3whenN=3.Forarbitraryn,heobtainin[17]thesameresultinthephasespaceH1locRNL2locRN.In2001,S.V.Zelik[53]consideredthenonautonomouscaseforequation.1,inwhichtheforcingtermgdependsontime.HeobtainedtheexistenceoflocallycompactglobalattractorandtheupperandlowerboundsfortheirKolomogorov's"-entropy.Someotherauthorshavealsoconsidereddierenttypesofwaveequationsinunboundeddomains[24],[25]inweightedspaces.InthischapterweestablishtheexistenceofglobalattractorsintheusualHilbertspacesH1L2,forequations.1and.2inunboundeddomainsofRN.Tothisendwecannot14

PAGE 21

applythesameprocedureasforboundeddomains,sincetheSobolevembeddingsarenolongercompact.Wewillapplythe"tailestimation"method,introducedfortherstorderlatticesystemsandthereactiondiusionequations[4,49,50].Itfeaturesan"approxima-tion"ofRNbysucientlylargeboundeddomainsk,thenusingthecompactnessoftheembeddingsinkandshowingtheuniformnullconvergenceofthesolutionsonRN)]TJ/F15 10.909 Tf 11.046 0 Td[(k,wenallyarrivetogettheasymptoticalcompactnessofthesemiow.2.1TheWaveEquationwithMassTermWeconsiderinthissection,thenonlinearwaveequationwithmassterm,utt+ut)-222(4u+u+fu=gx;t>0inRN.Weshallestablishrsttheexistenceandboundednessofsolutions,thenweshallprovetheasymptoticcompactnessofthecorrespondingsemiowtoobtaintheglobalat-tractor.2.1.1ExistenceofSolutionsandAbsorbingSetWestartbytransformingourproblemintoanabstractODEinthespaceL2H1andprovethatthenewoperatorismaximalaccretive.Thiswillallowustoshowtheexistenceofsolutionsandtheuniformboundednessofsuchsolutions.Weconsiderthesystemutt+ut)-222(4u+u+fu=gx;x2RN;t>0.3withinitialconditionsu;x=u0x;ut;x=u1xx2RN.4where>0,g2L2RN,andf2C1R;Rsatisesthefollowingcondition:f=0;fssFs0;8s2R.515

PAGE 22

whereisapositiveconstantandFs=Zs0ftdt.Inadditionweassumethat0limsups!1fs s<1.6NowsetH=L2RN,V=H1RN,andX=VHwiththeusualnormsandscalarproducts.WedenetheoperatorGinXby:DG=H2RNH1RNGw=0BBB@u)]TJ/F24 10.909 Tf 10.909 0 Td[(v)]TJ/F15 10.909 Tf 8.485 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(v+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1u1CCCA.7forw=u;v2DG.Then.3,.4areequivalenttotheinitialvalueprobleminX:8>>><>>>:wt+Gw=Rw;t>0;w2Xw=w0=u0;v0+u.8whereRw=0BBB@0)]TJ/F24 10.909 Tf 8.485 0 Td[(fu+g1CCCAThenextresultestablishesthemaximalaccretivityofthetheoperatorGinX.Lemma2.1.1Forasuitablechosentobe= 2+4,theoperatorGdenedpreviouslyismaximalaccretiveinX,andthereexistsaconstantC>0dependingonsuchthathGw;wiXCkwk2X;8w2DG.916

PAGE 23

Proof:Werstprovethepositivity.Letw=u;v2DG,thenhGw;wiX=hu)]TJ/F24 10.909 Tf 10.909 0 Td[(v;uiV+h)]TJ/F15 10.909 Tf 12.727 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(v+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1u;viH=kuk2V)]TJ/F29 10.909 Tf 10.909 8.788 Td[(RRNrurvdx)-222(hu;viH+RRNrurvdx+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvkH+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1hu;viH=kuk2V+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2H+2)]TJ/F24 10.909 Tf 10.909 0 Td[(hu;viHThensetting= p 2+4+p 2+4;.10wehavehGw;wiX)]TJ/F24 10.909 Tf 10.909 0 Td[(kuk2V+kvk2H)]TJ/F25 7.97 Tf 12.104 4.295 Td[( 2kvk2H)]TJ/F24 10.909 Tf 10.909 0 Td[(kuk2V+ 2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2H)]TJ/F24 10.909 Tf 8.485 0 Td[(kukVkvkH2q )]TJ/F24 10.909 Tf 10.909 0 Td[( 2)]TJ/F24 10.909 Tf 10.91 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(kukVkvkH)]TJ/F24 10.909 Tf 8.485 0 Td[(kukVkvkH:Wecancheckthat4)]TJ/F24 10.909 Tf 10.909 0 Td[( 2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(=22sothathGw;wiX)]TJ/F24 10.909 Tf 10.909 0 Td[(kwk2X)]TJ/F24 10.909 Tf 12.105 7.38 Td[( 2kvk20:ItsucestotakeC=NowweprovethattherangeofG+IequalsX.Letf=h;g2X;thequestioniswhetherthereexistsaw=u;v2DGsuchthat:Gw+w=f?17

PAGE 24

i.e.8<:u)]TJ/F24 10.909 Tf 10.909 0 Td[(v+u=h)]TJ/F15 10.909 Tf 8.485 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(v+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1u=gi.e.8<:v=+1u)]TJ/F24 10.909 Tf 10.909 0 Td[(h)]TJ/F15 10.909 Tf 8.485 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[([+1u)]TJ/F24 10.909 Tf 10.909 0 Td[(h]+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1u=gi.e.8<:v=+1u)]TJ/F24 10.909 Tf 10.909 0 Td[(h)]TJ/F15 10.909 Tf 8.485 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(+1u=g+)]TJ/F24 10.909 Tf 10.909 0 Td[(hNotethattheoperatorAu=)]TJ/F15 10.909 Tf 8.484 0 Td[(uinL2RNwithdomainH2RNisasectorialoperatorandthereexists!2Rsuchthat%Af2C:Re!gthisisaparticularcaseinExample1.2.1.Sotheequation)]TJ/F15 10.909 Tf 8.484 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(+1u=g+)]TJ/F24 10.909 Tf 10.909 0 Td[(hhasauniquesolutionu2H2RN,thuslettingv=+1u)]TJ/F24 10.909 Tf 11.08 0 Td[(handw=u;v,wegetauniquew2DGsuchthatGw+w=f.SotherangeofG+IequalsX.This,with.9,showsthatGismaximalaccretiveandnishestheproofoflemma2.1.1.Lemma2.1.1togetherwiththeLumer-PhillipsTheorem1.2.1implythat)]TJ/F24 10.909 Tf 8.484 0 Td[(GgeneratesanonexpansiveC0-semigroupe)]TJ/F25 7.97 Tf 6.587 0 Td[(GtonX.Furthermoresincefveries.6,theoperatorR:X!XislocallyLipschitzcontinuous.BythestandardtheoryofevolutionaryequationsseeG.R.Sell&Y.You[42],Theorem46.1thisleadstotheexistenceanduniquenessoflocalsolutionsasstatedinthenextlemma.Lemma2.1.2Ifg2L2RNandfsatises.6,thenforanyinitialdataw0=u0;v02X,thereexistsauniquelocalsolutionwt=ut;vtof.8suchthatw2C1)]TJ/F24 10.909 Tf 8.485 0 Td[(T0;T0;EforsomeT0=T0w0>0.Infactwewillshowthatthelocalsolutionwtof.8isboundedandexistsglobally.Lemma2.1.3Assumethat2.5and.6aresatisedandthatg2H.Thenanysolution18

PAGE 25

wtofproblem.8satiseskwtkXM;tT1.11whereMisaconstantdependingonlyon;gandT1dependingonthedata;g;Rwhenkw0kXR.Proof:Letw02DGbetheinitialconditionin.8.Takingtheinner-productof.8withwinXwendthat1 2d dtkwk2X=hGw;wiX+hRw;wiE=hGw;wiX+hg;viH)-222(hf;viH)]TJ/F24 10.909 Tf 26.933 0 Td[(Ckwk2X+kgkHkvkH)]TJ/F24 10.909 Tf 10.909 0 Td[(hfu;uiH)-222(hfu;utiH;by.5wehave)]TJ/F24 10.909 Tf 8.484 0 Td[(hfu;uiH)]TJ/F24 10.909 Tf 20 0 Td[(ZRNFudxandhfu;utiH=)]TJ/F24 10.909 Tf 11.651 7.38 Td[(d dtZRNFudx:ThenusingtheYounginequality,itfollowsforany>0that1 2d dtkwk2X)]TJ/F24 10.909 Tf 21.818 0 Td[(Ckwk2X+ 2kvk2H+1 2kgk2H)]TJ/F24 10.909 Tf 10.909 0 Td[(ZRNFudx)]TJ/F24 10.909 Tf 14.074 7.38 Td[(d dtZRNFudxwhichimpliesthatd dtkwk2X+2ZRNFudx2)]TJ/F24 10.909 Tf 10.91 0 Td[(Ckwk2X)]TJ/F15 10.909 Tf 10.909 0 Td[(2ZRNFudx+1 kgk2HNowwecanchoosesmallenoughsothat)]TJ/F24 10.909 Tf 10.909 0 Td[(C<0andtaking19

PAGE 26

=minf)]TJ/F15 10.909 Tf 13.939 0 Td[(2)]TJ/F24 10.909 Tf 10.909 0 Td[(C;g>0wehaved dtkwk2X+2ZRNFudx)]TJ/F24 10.909 Tf 20 0 Td[(kwk2X+2ZRNFudx+1 kgk2H.12andthenbyUniformGronwallinequalitywegetkwk2X+2ZRNFudxe)]TJ/F25 7.97 Tf 6.586 0 Td[(tkw0k2X+2ZRNFu0dx+)]TJ/F24 10.909 Tf 10.909 0 Td[(e)]TJ/F25 7.97 Tf 6.586 0 Td[(t1 kgk2Hwhichyieldskwk2Xe)]TJ/F25 7.97 Tf 6.586 0 Td[(tkw0k2X+2ZRNFu0dx+1 kgk2H:.13Nowby.5wehaveZRNFu0dx1 ZRNfu0u0dxC ZRNu20xdx:Thenwededucefrom.13thatforeveryw02DG,kwk2Xe)]TJ/F25 7.97 Tf 6.586 0 Td[(tkw0k2X+C ku0k2H+1 kgk2H:.14AndbydensityofDGinXandthecontinuityofthesolutionof.8inX;Tw0weseethat2.13holdsforeveryw02X.NowletR>0andkw0kXR,thenku0kHRandkwk2Xe)]TJ/F25 7.97 Tf 6.586 0 Td[(tR2+CR2 +1 kgk2H.15whichyieldskwk2X2 kgk2H;fortT1=1 lnR2+CR2 kgk2H.16and.11followswithM=2 kgk2Handtheproofiscomplete.By.15,Wehavealsothefollowingresult.20

PAGE 27

Lemma2.1.4Letg2H.ThenforanygivenT>0,everysolutionwof2.8satiseskwkXL;0tT.17whereLdependson;;kgkH,Tandkw0kX.Lemma2.1.3impliesthatthesolutionwtexistsglobally,thatisTw0=+1,whichimpliesthatthesystem.8generatesacontinuoussemiowfStgt0onX.DenotebyOtheballO=fw2X:kwkXMg.18whereMistheconstantin.11.Thenitfollowsfrom.11thatOisanabsorbingsetforStinXandthatforeveryboundedsetBinXthereexistsaconstantTBdependingonlyon;gandBsuchthatStBO;tTB:.19InparticularthereexistsaconstantT0dependingonlyon;gandOsuchthatStOO;tT0:.202.1.2GlobalAttractorTheexistenceofanabsorbingsetistherststeptowardtheexistenceofaglobalattractor.WeneednowtoprovetheasymptoticcompactnessofSt.ThekeyidealiesinestablishinguniformestimatesonTailEnds"ofsolutions,thatis,thenormofthesolutionswtareuniformlysmallwithrespecttotoutsideasucientlylargeball.Lemma2.1.5If.5and.6hold,g2Handw0=u0;v02O,thenforevery">0,thereexistspositiveconstantsT"andK"suchthatthesolutionwt=ut;vtofproblem.8satisesZjxjknjuj2+jruj2+jvj2odx";tT";kK":.2121

PAGE 28

Proof:Chooseasmoothfunctionsuchthat0s1fors2R+,ands=0for0s1;s=1fors2:ThenthereexistsaconstantC>0suchthatj0sjCfors2R+.Letwt=ut;vtbethesolutionofproblem.8withinitialconditionw0=u0;v02Othenvt=u+utsatisestheequationvt)]TJ/F15 10.909 Tf 10.909 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(v+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1u=)]TJ/F24 10.909 Tf 8.485 0 Td[(fu+g.22takinginnerproductof.22withjxj2 k2vinHwegetZRNjxj2 k2vvtdx)]TJ/F29 10.909 Tf 10.909 14.848 Td[(ZRNujxj2 k2vdx+)]TJ/F24 10.909 Tf 10.909 0 Td[(ZRNjxj2 k2jvj2dx+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1ZRNjxj2 k2uvdx=)]TJ/F29 10.909 Tf 10.303 14.849 Td[(ZRNfujxj2 k2vdx+ZRNjxj2 k2gvdx.23But)]TJ/F29 10.909 Tf 10.303 14.848 Td[(ZRNujxj2 k2vdx=ZRNjxj2 k2rurv+2 k2ZRN0jxj2 k2vxru=ZRNjxj2 k2jruj2+rurut+2 k2ZRN0jxj2 k2vxru=1 2d dtZRNjxj2 k2jruj2+ZRNjxj2 k2jruj2+2 k2ZRN0jxj2 k2vxru;and2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1ZRNjxj2 k2uvdx=2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1ZRNjxj2 k2juj2+uut=1 22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1d dtZRNjxj2 k2juj2+2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1ZRNjxj2 k2juj2:22

PAGE 29

Then.23becomes1 2d dtZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2+ZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2+)]TJ/F15 10.909 Tf 10.909 0 Td[(2ZRNjxj2 k2jvj2=)]TJ/F29 10.909 Tf 10.303 14.848 Td[(ZRNjxj2 k2fuu+ut+ZRNjxj2 k2gvdx)]TJ/F15 10.909 Tf 14.755 7.38 Td[(2 k2ZRN0jxj2 k2vxru;.24andsinceZRNjxj2 k2fuu+utd dtZRNjxj2 k2Fu+ZRNjxj2 k2Fu;wededucethatd dtZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2+2Fu+ZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2+2Fu)]TJ/F15 10.909 Tf 20 0 Td[()]TJ/F15 10.909 Tf 10.909 0 Td[(2ZRNjxj2 k2jvj2+ZRNjxj2 k2gvdx)]TJ/F15 10.909 Tf 14.755 7.38 Td[(2 k2ZRN0jxj2 k2vxru;.25where=minf1;g.Now,thereexistsaconstantK">0suchthatforkK,wehave)]TJ/F15 10.909 Tf 8.485 0 Td[()]TJ/F15 10.909 Tf 10.909 0 Td[(2ZRNjxj2 k2jvj2+ZRNjxj2 k2gvdx)]TJ/F15 10.909 Tf 14.755 7.38 Td[(2 k2ZRN0jxj2 k2vxru" 2;whichimpliesbyUniformGronwallinequalitythat23

PAGE 30

ZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2+2Fue)]TJ/F25 7.97 Tf 6.587 0 Td[(tZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1ju0j2+jru0j2+jv0j2+2Fu0+"1)]TJ/F24 10.909 Tf 10.909 0 Td[(e)]TJ/F25 7.97 Tf 6.587 0 Td[( 2:Nowsincew02O,thereexistaconstantM>0,uniformlychosenforw02O,suchthatZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1ju0j2+jru0j2+jv0j2+2Fu0M:ThenwegetforkK",ZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2+2FuMe)]TJ/F25 7.97 Tf 6.587 0 Td[(t+"1)]TJ/F24 10.909 Tf 10.91 0 Td[(e)]TJ/F25 7.97 Tf 6.587 0 Td[( 2:ChoosingT"=1 ln2M 2")]TJ/F24 10.909 Tf 10.909 0 Td[(",wededucethatZRNjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2"fortT";kK"whichyields.21since0<2)]TJ/F24 10.909 Tf 10.67 0 Td[(+1<1fortheparticularchoiceof,andtheproofiscomplete.Bymultiplyingequation.22withvandintegratingwededucethefollowingenergyequationd dtEwt+2Ewt=Gwt8t>0;.26whereEwisthequasi-energyfunctional,Ew=2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1kuk2H+kruk2L2RN+kvk2H;.2724

PAGE 31

andGw=)]TJ/F15 10.909 Tf 8.484 0 Td[(2)]TJ/F15 10.909 Tf 10.909 0 Td[(2kvk2H+2ZRNgvdx)]TJ/F15 10.909 Tf 10.909 0 Td[(2ZRNfuvdx:.28ThisenergyfunctionalEwillbeusedlaterasanequivalentnorm,moresuitableinprovingtheasymptoticalcompactness.Thefollowinglemmawillbealsousefulinprovingtheasymptoticalcompactness.Lemma2.1.6Letwn=un;vn)167(!w0=u0;v0weaklyinX,thenforeveryT>0wehaveStwn)167(!Stw0weaklyinL2;T;X.29andStwn)167(!Stw0weaklyinX;for0tT:.30Proof:SincefwngnconvergesweaklyinX,thenitisboundedinXsothat,bylemma2.1.4fStwngnisboundedinL1;T;X.This,with.8,impliesthat@ @tStvnisboundedinL1;T;H)]TJ/F22 7.97 Tf 6.586 0 Td[(1RN2.31andStvnisboundedinL1;T;L2RN:.32Weinferthatthereexistsasubsequencefwnjgjandw1=u1;v12L1;T;XsuchthatStwnj)167(!w1weaklyinL2;T;X;.33@ @tStvnj)167(!@ @tv1weaklyinL1;T;H)]TJ/F22 7.97 Tf 6.586 0 Td[(1RN.34and@ @tStunj)167(!@ @tu1weaklyinL2;T;H:.3525

PAGE 32

Wecanshowthatw1isasolutionof.8withw1=w0.Indeed,wehavebythemildsolutionformula,Stwnj=e)]TJ/F25 7.97 Tf 6.587 0 Td[(Gtwnj+Zt0e)]TJ/F25 7.97 Tf 6.586 0 Td[(Gt)]TJ/F25 7.97 Tf 6.587 0 Td[(sRSswnjds:.36And,sincewnj!w0weaklyinX,wededucefrom.33thate)]TJ/F25 7.97 Tf 6.586 0 Td[(Gtwnj+Zt0e)]TJ/F25 7.97 Tf 6.587 0 Td[(Gt)]TJ/F25 7.97 Tf 6.586 0 Td[(sRSswnjds)167(!e)]TJ/F25 7.97 Tf 6.587 0 Td[(Gtw0+Zt0e)]TJ/F25 7.97 Tf 6.587 0 Td[(Gt)]TJ/F25 7.97 Tf 6.586 0 Td[(sRw1sds;.37weaklyinX.whichimplies,bytheuniquenessofweaklimitthatw1t=e)]TJ/F25 7.97 Tf 6.587 0 Td[(Gtw0+Zt0e)]TJ/F25 7.97 Tf 6.587 0 Td[(Gt)]TJ/F25 7.97 Tf 6.586 0 Td[(sRw1sds:.38Thatisw1isasolutionof.8andbytheuniquenessofsolutionswehavew1t=Stw0.ThisshowsthatanysubsequenceofStwnhasaweaklyconvergentsubsequenceinL2;T;X,thereforeweconclude.29.Asimilarargumentyields.30.Similarto2.29wealsohavethatifwn)167(!wweaklyinX,thenfor0sT,Stwn)167(!Stw0weaklyinL2s;T;X.39Westatehereanotherusefullemma.Lemma2.1.7LetbeaboundeddomaininRN.Supposeun)167(!uinL2andvn)167(!vweaklyinL2,thenZfunvndx)167(!ZfuvdxinRuptoasubsequenceextraction.Proof:By.6wecanshow,uptoasubsequence,thatfun)167(!fuinL2.NowdenethelinearfunctionalsInandIonL2byInv=Zfunvdx;Iv=Zfuvdx:ThenIn)167(!IinL2thedualspaceofL2.IndeedjInv)]TJ/F24 10.909 Tf 10.909 0 Td[(IvjZjfun)]TJ/F24 10.909 Tf 10.909 0 Td[(fujjvjdx26

PAGE 33

kfun)]TJ/F24 10.909 Tf 10.909 0 Td[(fukL2kvkL2:whichimpliesthatkIn)]TJ/F24 10.909 Tf 10.909 0 Td[(IkL2kfun)]TJ/F24 10.909 Tf 10.909 0 Td[(fukL2)167(!0asn)167(!1:SoIn!IinL2andvn!vweaklyinL2,thenitfollows,byaclassicalresultinfunctionalanalysisthatInvn)167(!Iv,whichprovesthelemma.WearenowreadytoprovetheasymptoticcompactnessofthesemiowSt.Theorem2.1.1ThesemiowStgeneratedbythesystem.8isasymptoticallycompactinX,thatisiffwngnisaboundedsequenceinXandtn)167(!+1,thenfStwngn1isprecompactinX.Proof:LetwnbeaboundedsequenceinXwithkwnkXRandtn)167(!+1thenby.19thereexistsaconstantTR>0dependingonlyonR>0suchthatStwn2O;8n1;8tTR:.40Sincetn)167(!+1,thereexistsN1RsuchthatnN1impliestnTRsothatStnwn2O;8nN1R:.41Thenthereexistsw2Xsuchthat,uptoasubsequenceStnwn)167(!wweaklyinX:.42NowforeveryT>0thereexistsN2R;TsuchthatfornN2R;Twehavetn)]TJ/F24 10.909 Tf 8.871 0 Td[(TTRsothatStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twn2O8nN2R;T:.43ThusthereisawT2OsuchthatStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twn)167(!wTweaklyinX;.4427

PAGE 34

andbytheweakcontinuity.30wemusthavew=STwTwhichimpliesthatliminfn!1kStnwnkXkwkX:.45Soweonlyneedtoprovethatlimsupn!1kStnwnkXkwkX:.46Bytheenergyequation.26,itfollowsthatanysolutionwt=Stwof.8satisesEStw=e)]TJ/F22 7.97 Tf 6.586 0 Td[(2t)]TJ/F25 7.97 Tf 6.586 0 Td[(sESsw+Ztse)]TJ/F22 7.97 Tf 6.586 0 Td[(2t)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrwdr;ts0:.47whereEandGaregivenby2.27and.28,respectively.Inthefollowing,T0istheconstantin.20,andfor">0,T"istheconstantin.21.LetT0"beaxedconstantsuchthatT0"maxfT";T0g.TakingTT0",andapplying.47tothesolutionStStn)]TJ/F24 10.909 Tf 10.767 0 Td[(Twnwiths=T0andt=T,thenweget,fornN2R;T,EStnwn=ESTStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twn=e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(T0EST0Stn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twn+ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twndr:.48SinceT0T0wehaveST0Stn)]TJ/F24 10.909 Tf 9.473 0 Td[(Twn2OfornN2R;T,thereforebythedenitionofEwendthate)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0EST0Stn)]TJ/F24 10.909 Tf 10.909 0 Td[(TwnCe)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0;8nN2R;T:.49Ontheotherhand,wehaveZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twndr28

PAGE 35

=)]TJ/F15 10.909 Tf 8.484 0 Td[(2)]TJ/F24 10.909 Tf 10.909 0 Td[(ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rkSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvnk2dr+2ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZRNgSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvndxdr)]TJ/F15 10.909 Tf 8.484 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZRNfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvndxdr.50Let'shandletherstandlasttermof.50.Sincewehave,e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvn)167(!e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rSrvweaklyinL2T0;T;H;itfollowsthat:liminfn!1ke)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TvnkL2T0;T;Hke)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rSrvkL2T0;T;H;whichimpliesthatlimsupn!1)]TJ/F15 10.909 Tf 8.485 0 Td[(2)]TJ/F24 10.909 Tf 10.909 0 Td[(ke)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TvnkL2T0;T;H)]TJ/F15 10.909 Tf 20 0 Td[(2)]TJ/F24 10.909 Tf 10.909 0 Td[(ke)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rSrvkL2T0;T;H:.51Alsoby.44and.39wehaveZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZRNgSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvndxdr)167(!ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZRNgSrvTdxdr.52Nowlet'shandlethenonlineartermof.50.Wehave)]TJ/F15 10.909 Tf 8.485 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZRNfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.91 0 Td[(Tvndxdr=)]TJ/F15 10.909 Tf 8.485 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZjxjkfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.91 0 Td[(Tvndxdr)]TJ/F15 10.909 Tf 8.485 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZjxjkfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvndxdr:.5329

PAGE 36

Handlingthersttermontheright-handsideof2.53gives2ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZjxjkfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TvndxdrCZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZjxjkjSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunjjSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TvnjCZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZjxjkjSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tunj2!1 2ZjxjkjSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvnj2!1 2"2CZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rdr"2C 2;nN2R;T:.54Wetreatnowthesecondtermontheright-handsideof.53.Wewanttoprovethatasn!+1,ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZjxjkfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvndxdr)167(!ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZjxjkfSruTSrvTdxdr.55Setk=fx2RN:jxjkgandletr2[T0;T].ThenwehaveSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twn)167(!SrwT;weaklyinX:BythecompactnessoftheSobolevembeddingH1kL2k,weinferthatSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tun)167(!SruT;stronglyinL2k.56andSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvn)167(!SrvT;weaklyinL2k2.57then.55followsfromlemma2.1.7.30

PAGE 37

By.53,.54and.55wendthatforkK",limsupn!1)]TJ/F15 10.909 Tf 8.485 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZRNfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvndxdr"C)]TJ/F15 10.909 Tf 10.909 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZjxjkfSruTSrvTdxdr:Lettingk)167(!1weobtainlimsupn!1)]TJ/F15 10.909 Tf 8.485 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZRNfSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TunSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Tvndxdr"C)]TJ/F15 10.909 Tf 10.909 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZRNfSruTSrvTdxdr:.58By.50,.51,.52and.58,wenallyobtainlimsupn!1ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(Twndr)]TJ/F15 10.909 Tf 20 0 Td[(2)]TJ/F24 10.909 Tf 10.909 0 Td[(ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rkSrvTk2dr+2ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(rZRNgSrvTdxdr)]TJ/F15 10.909 Tf 8.484 0 Td[(2ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rZRNfSruTSrvTdxdr+"C;thatislimsupn!1ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TwndrZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrwTdr+"C:.59Takinglimitof.48,.49and.59weget,asn!1,limsupn!1EStnwnCe)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(T0+ZTT0e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrwTdr+"C:.60Ontheotherhand,sincew=STwT,by.47wealsohavethatEw=ESTwT=e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0EST0wT+ZTT0e)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.586 0 Td[(rGSrwTdr:.6131

PAGE 38

Henceitfollowsfrom2.60-.61thatlimsupn!1EStnwnEw+Ce)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0+"C)]TJ/F24 10.909 Tf 10.909 0 Td[(e)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0EST0wT:.62NowsincewT2OandT0TOwendthatje)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0EST0wTjCe)]TJ/F22 7.97 Tf 6.587 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0:Thenfrom.62wehavelimsupn!1EStnwnEw+Ce)]TJ/F22 7.97 Tf 6.586 0 Td[(2T)]TJ/F25 7.97 Tf 6.587 0 Td[(T0+"C:.63Nowtakinglimitof.63asT!1andthenletting"!0,weobtainlimsupn!1EStnwnEw;thatislimsupn!12)]TJ/F24 10.909 Tf 10.909 0 Td[(+1kStnunk2H+krStnunk2L2RN+kStnvnk2H2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1kuk2H+kruk2L2RN+kvk2H:.64NotingthatEw=2)]TJ/F24 10.909 Tf 11.042 0 Td[(+1kuk2H+kruk2L2RN+kvk2HisequivalenttothenormofX,wecanassumewithoutlossofgeneralitythatthenormofXisdenedbyit.Thenwehavelimsupn!1kStnwnkXkwkXasdesiredin.46.ThereforewegetthestrongconvergenceofStnwntowinX.Theproofiscomplete.Nowwestateourmainresultobtainedinthissection.Theorem2.1.2Assumethatfsatises.5,2.6andg2L2RN.Then,problem.8possessesaglobalattractorinX=H1RNL2RNwhichisacompactinvariantsubsetthatattractseveryboundedsetofXwithrespecttothenormtopology.32

PAGE 39

Proof:Sincewehaveestablishedtheexistenceofanabsorbingsetin.18andtheasymp-toticcompactnessofthesemiowStinXinTheorem2.1.1,theconclusionfollowsfromTheorem1.1.1.2.2TheWaveEquationWithoutMassTermInthissectionwewillstudytheexistenceofglobalattractorforthewaveequationwithoutmassterm,8>>><>>>:utt+ut)-222(4u+fu=0;x2;t>0;uj@=0;u;x=u0x;ut;x=u1x.65whereisadomainofRNboundedonlyinonedirection,withsmoothboundary.Thecase=RN,forthisequationisstillanopenproblemduetosomedicultiesingettinganinequalitysuchas.9fortheoperatorGinH1norm.InourcasewewilluseanequivalentnormprovidedbythePoincareinequalityforwhichthedesiredestimateworks.Weassumethesamecondtions.5and.6forthenonlinearfunctionf.WewillworkinthephasespaceX=VHwhereV=H10;H=L2.HisendowedwiththenormandinnerproductforL2andVisendowedwiththeinnerproductandnormdenedasfollows,u;vV=Zrurvdx;u;v2VandkukV=krukHn;u2V:.66NowdenethefollowingbilinearoperatorinV:u;v1=Zuvdx+Zrurvdx;u;v2V;.67whichisalsoaninnerproductinVwithinducednormkuk1=hkuk2H+kruk2Hni1 2.BythePoincareinequalitykkVandkk1areequivalentnormsinV.Thatis,therearepositive33

PAGE 40

constantsC1andC2suchthatC1kukVkuk1C2kukV;8u2V:.68Let'smakeatransformationtowritetheequation.65asarstorderabstractODE.Choose= 2+4andsetv=u+ut,w=0@uv1A.Then,problem.65isequivalentto8<:wt+Gw=Rw;t>0;w2Xw=w0=u0;u1+u0.69whereRw=0@0)]TJ/F24 10.909 Tf 8.485 0 Td[(fu1AandGw=0@u)]TJ/F24 10.909 Tf 10.909 0 Td[(v)]TJ/F15 10.909 Tf 8.485 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(v+2)]TJ/F24 10.909 Tf 10.909 0 Td[(u1Aforw=0@uv1A2DG=H2H10H1.AsinLemma2.1.1weshowthepositivityoftheoperatorGwithasimilarestimate.Lemma2.2.1For= 2+4,theoperatorGismaximalaccretiveinXandveriesthefollowingGw;wXkwk2X+ 2kvk2H;8w=0@uv1A2X;.70where= p 2+4+p 2+4:.7134

PAGE 41

Proof:Letw=0@uv1A2Xthenwehave:Gw;wX=u)]TJ/F24 10.909 Tf 10.909 0 Td[(v;uV+)]TJ/F15 10.909 Tf 8.485 0 Td[(u+)]TJ/F24 10.909 Tf 10.91 0 Td[(v)]TJ/F24 10.909 Tf 10.909 0 Td[(u;vH=kuk2V)]TJ/F15 10.909 Tf 10.909 0 Td[(ru;rvHn+)]TJ/F15 10.909 Tf 8.485 0 Td[(u;vH+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2H)]TJ/F24 10.909 Tf 8.485 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(u;vH=kuk2V+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2H)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.91 0 Td[(u;vHkuk2V+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2H)]TJ/F24 10.909 Tf 10.909 0 Td[(kukHkvkH:Thensetting= p 2+4+p 2+4asin.10,wehaveGw;wX)]TJ/F24 10.909 Tf 10.909 0 Td[(kuk2V+kvk2H)]TJ/F25 7.97 Tf 12.104 4.295 Td[( 2kvk2H)]TJ/F24 10.909 Tf 10.909 0 Td[(kuk2V+ 2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2H)]TJ/F24 10.909 Tf 8.485 0 Td[(kukVkvkH2q )]TJ/F24 10.909 Tf 10.909 0 Td[( 2)]TJ/F24 10.909 Tf 10.91 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(kukVkvkH)]TJ/F24 10.909 Tf 8.485 0 Td[(kukVkvkHwecancheckthat4)]TJ/F24 10.909 Tf 10.91 0 Td[( 2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(=22sothatGw;wX)]TJ/F24 10.909 Tf 10.909 0 Td[(kwk2X)]TJ/F24 10.909 Tf 12.105 7.38 Td[( 2kvk2H0:Theproofiscomplete.Theexistenceofsolutionfor.69followsinthesameapproachasforequation.8.Similarly,wecanproveananalogousresultasinlemma2.1.3andwehaveshownthatthere35

PAGE 42

alsoexistsaboundedabsorbingsetOinX.Nowlet'sestablishthetailendsestimatesforequation.69.Lemma2.2.2If.5,.6hold,g2Handw0=u0;v02O,thenforevery">0,thereexistsT"andK"suchthatthesolutionwt=ut;vtofproblem.69satisesZfjxjkghjutj2+jrutj2+jvtj2idx";tT";kK":.72Proof:Theproofworksbasicallylikethatforequation.8.Anysolutionwt=0@utvt1Asatises:vt)]TJ/F15 10.909 Tf 10.909 0 Td[(u+)]TJ/F24 10.909 Tf 10.909 0 Td[(v+2)]TJ/F24 10.909 Tf 10.909 0 Td[(u=)]TJ/F24 10.909 Tf 8.485 0 Td[(fu+g.73andut+u=v:.74Wechoosethesamecut-ofunction.NowtakeinnerproductinHofjxj2 k2vxwith.73togetZjxj2 k2vvtdx)]TJ/F29 10.909 Tf 10.909 14.849 Td[(Zujxj2 k2vdx+)]TJ/F24 10.909 Tf 10.909 0 Td[(Zjxj2 k2jvj2dx+2)]TJ/F24 10.909 Tf 10.909 0 Td[(Zjxj2 k2uvdx=)]TJ/F29 10.909 Tf 10.303 14.849 Td[(Zfujxj2 k2vdx+Zjxj2 k2gvdx:.75But)]TJ/F29 10.909 Tf 10.303 14.849 Td[(Zujxj2 k2vdx=Zjxj2 k2rurv+2 k2Z0jxj2 k2vxru=Zjxj2 k2jruj2+rurut+2 k2Z0jxj2 k2vxru=1 2d dtZjxj2 k2jruj2+Zjxj2 k2jruj2+2 k2Z0jxj2 k2vxru;36

PAGE 43

and2)]TJ/F24 10.909 Tf 10.909 0 Td[(Zjxj2 k2uvdx=2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1Zjxj2 k2juj2+uut=1 22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1d dtZjxj2 k2juj2+2)]TJ/F24 10.909 Tf 10.91 0 Td[(+1Zjxj2 k2juj2:Then.73becomes1 2d dtZjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(juj2+jruj2+jvj2+Zjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(juj2+jruj2+jvj2+)]TJ/F15 10.909 Tf 10.909 0 Td[(2Zjxj2 k2jvj2=)]TJ/F29 10.909 Tf 10.303 14.848 Td[(Zjxj2 k2fuu+ut+Zjxj2 k2gvdx)]TJ/F15 10.909 Tf 14.755 7.38 Td[(2 k2Z0jxj2 k2vxru:.76But2)]TJ/F24 10.909 Tf 10.665 0 Td[(couldbenegativeforcertainvaluesof.Since2)]TJ/F24 10.909 Tf 10.665 0 Td[(+1>0,let'sintroduceanotherequationtogetamoredesirableidentity.Takinginnerproductofjxj2 k2uxwith.74,weget1 2d dtZjxj2 k2juj2dx+Zjxj2 k2juj2dx=Zjxj2 k2uvdx:Andaddingtheaboveand.76yields1 2d dtZjxj2 k22)]TJ/F24 10.909 Tf 10.91 0 Td[(+1juj2+jruj2+jvj2+Zjxj2 k22)]TJ/F24 10.909 Tf 10.909 0 Td[(+1juj2+jruj2+jvj2+)]TJ/F15 10.909 Tf 10.909 0 Td[(3Zjxj2 k2jvj2=)]TJ/F29 10.909 Tf 10.303 14.849 Td[(Zjxj2 k2fuu+ut+Zjxj2 k2gvdx)]TJ/F15 10.909 Tf 14.755 7.38 Td[(2 k2Z0jxj2 k2vxru:.7737

PAGE 44

Thentheconclusionfollowsthesamewayasintheproofoflemma2.1.5.Similarly,wehavethefollowingenergyequationforthesolutionof.69,d dtEwt+2Ewt=Gwt8t>0;.78whereEw=2)]TJ/F24 10.909 Tf 10.909 0 Td[(+1kuk2H+kruk2HN+kvk2H;.79andGw=)]TJ/F15 10.909 Tf 8.485 0 Td[(2)]TJ/F15 10.909 Tf 10.909 0 Td[(3kvk2H+2Zgvdx)]TJ/F15 10.909 Tf 10.909 0 Td[(2Zfuvdx:.80Therestoftheproofofexistenceofaglobalattractorisagainsimilartothecasewithmassterm.WegetthemainresultinthissectionTheorem2.2.1LetbeadomainofRNboundedinonlyonedirection.Assumethatfsatises.5,.6andg2L2.Then,problem.69possessesaglobalattractorinX=H10L2whichisacompactinvariantsubsetthatattractseveryboundedsetofXwithrespecttothenormtopology.38

PAGE 45

3DYNAMICSOFSECONDORDERLATTICESYSTEMSInthischapterwetakeonthelongtimedynamicsofasecondorderlatticedierentialequationLDE.BroadlyspeakinganLDEisaninnitesystemofordinarydierentialequationswithadiscretestructureinthephasespace.TheyoftencomefromaspatialdicretizationofanevolutionaryPDE.However,manyLDE'soccurasmodelsintheirownrightandarenotapproximationstothecontinuumlimit.Latticesystemsoccurinmanyapplicationssuchaselectriccircuittheory,neuralnetworks,materialscience,theoryofchemicalreactions,imageprocessing,andbiology.ThemathematicalstudyofLDE'sisquiterecent:theliteraturegoesbacktoabout1987,withthefullmathematicaldevelopmentstartingin1990's.Weconsiderinthischapterthefollowinglatticesystemui+_ui)]TJ/F15 10.909 Tf 10.909 0 Td[(ui)]TJ/F22 7.97 Tf 6.587 0 Td[(1)]TJ/F15 10.909 Tf 10.909 0 Td[(2ui+ui+1+fui=gi;i2Z.1where_uandurepresentrespectivelytherstandsecondderivativesofuwithrespecttotimet,fisanonlinearfunctionsatisfyingsomegrowthconditionsandg=gii2Z2`2.Equation3.1canbeviewedasaspatialdiscretizationoftheone-dimensionaldampednonlinearwaveequation,utt+ut)]TJ/F24 10.909 Tf 10.91 0 Td[(uxx+fu=g;x2R:.2Inthischapterwewillshowtheexistenceofglobalattractorforthesemiowgeneratedby.1.Hereagainthekeyliesinavariantofthetailendestimates"usedinthepreviouschapterforthenonlinearwaveequation.39

PAGE 46

3.1TheExistenceandBoundednessofSolutionsInthissectionweprovetheexistenceanduniquenessofsolutionsofthefollowingsecondorderlatticesystem,foralltimet0.Wealsoshowtheuniformboundednessofsolutions.Considerthesystemui+_ui)]TJ/F15 10.909 Tf 10.909 0 Td[(ui)]TJ/F22 7.97 Tf 6.587 0 Td[(1)]TJ/F15 10.909 Tf 10.909 0 Td[(2ui+ui+1+fui=gi;i2Z.3withinitialconditionsui=ui;0;_ui=ui;1;i2Z:.4Here>0isaconstantandf2C1R;Rsatisesf=0andthefollowingcondition:fssFs0;8s2R.5whereisapositiveconstantandFs=Zs0ftdt.Weremarkthatcondition.5issatisediffisanondecreasingfunctionsatisfyingfsso;forinstance,iffisapolyno-mialwithpositivecoecientsandodddegreemonomials.Wewillconsiderthespace`2=fu=uii2ZjXi2Zu2i<1gwhichisaHilbertspacewiththeusualinnerproductu;v=Xi2Zuiviandnormkuk=Xi2Zu2i1 2.IntroducetwolinearoperatorsB,BandAfrom`2to`2asfollows.Foru=uii2Z2`2,deneBui=ui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(ui;Bui=ui)]TJ/F22 7.97 Tf 6.587 0 Td[(1)]TJ/F24 10.909 Tf 10.909 0 Td[(ui;andAui=)]TJ/F15 10.909 Tf 8.485 0 Td[(ui)]TJ/F22 7.97 Tf 6.587 0 Td[(1)]TJ/F15 10.909 Tf 10.909 0 Td[(2ui+ui+1:.6ThenweseethatA=BB=BB=)]TJ/F15 10.909 Tf 8.485 0 Td[(B+B3.7Bu;v=u;Bv;andAu;v=Bu;Bv;8u;v2`2:.840

PAGE 47

Thebilinearformu;v1=Bu;Bvdenesalsoaninnerproductin`2withinducednormkuk1=kBuk.WeletH=`2;;;kk;V=`2;;1;kk1whichareHilbertspaces:Thenormskkandkk1areequivalentnormsin`2.Infactwehavekuk2kuk21=Xi2Zjui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(uij24kuk28u2`2:.93.1.1TheExistenceandUniquenessofSolutionsIntheremaininganalysisourphasespacewillbeX=VHequippedwiththeproducttopology,whichmakesXaHilbertspace.TheinnerproductandnorminXareasfollows:for'j=0@ujvj1A2Xj=1;2wehave'1;'2X=u1;u21+v1;v2;k'k2X=';;'X8'=0@uv1A2X:Letv=u+_u,where>0isapositiveparameterchosenas= 2+4:.10Thentheinitialvalueproblem.3,.4canbereformulatedasarstorderabstractODEinXasfollows_'+G'=R';'=0@u0v01A.1141

PAGE 48

whereGandRaredenedonXasfollows:G'=0BBB@u)]TJ/F24 10.909 Tf 10.909 0 Td[(vAu+)]TJ/F24 10.909 Tf 10.909 0 Td[(v)]TJ/F24 10.909 Tf 10.909 0 Td[(u1CCCAandR'=0@0)]TJ/F24 10.909 Tf 8.485 0 Td[(fu+g1A:Wemaketheabusivenotationsfu=fuii2Z,Fu=Fuii2Z.Nowletu2`2thensincef=0wehavekfuk2=Xi2Zjf0iuijjuij2;wherei2;1.Byjiuijjuijkuk,wegetkfukkukmaxkukskukf0s:.12Itfollowsfrom.12thatfu2`2.ThusRmapsXintoitself.Nextweprovetheexistenceanduniquenessofthesolutionof.11asstatedinthenextlemma.Lemma3.1.1Foreveryinitialdata'=0@u0v01A2X,thereisauniquelocalsolution't=0@utvt1Aof3.11suchthat'2C1[)]TJ/F24 10.909 Tf 8.485 0 Td[(T'0;T'0]forsomeT'0>0.Proof:Wejustneedtoprovethat'7!R')]TJ/F24 10.909 Tf 10.16 0 Td[(G'islocallyLispchitzfromXintoitself.LetBbeaboundedsubsetofXand'1;'22B,thensimilarto.12,thereexistsaconstantLBdependingonBsuchthatkR'1)]TJ/F24 10.909 Tf 10.909 0 Td[(R'2k2X=kfu1)]TJ/F24 10.909 Tf 10.909 0 Td[(fu2k2=Xi2Zjf0u1i+iu2i)]TJ/F24 10.909 Tf 10.909 0 Td[(u1ij2ju1i)]TJ/F24 10.909 Tf 10.909 0 Td[(u2ij2LBk'1)]TJ/F24 10.909 Tf 10.909 0 Td[('2k2X:42

PAGE 49

ThereforeRislocallyLipschitz.Ontheotherhand,itiseasytoseethatGisaboundedlinearoperatorsothatR')]TJ/F24 10.909 Tf 11.307 0 Td[(G'islocallyLipschitzfromXtoX.TheconclusionofthelemmafollowsfromthestandardtheoryofabstractordinarydierentialequationsinBanachspaces.3.1.2TheBoundednessofSolutionsWestartwithpresentingapositivityestimateforthelinearoperatorG,whichiscrucialtowardprovingtheexistenceofabsorbingset.Infactitisakeyestimateusedinthiswork.Lemma3.1.2TheoperatorGveries:G';'Xk'k2X+ 2kvk2;8'=0@uv1A2X;.13where= p 2+4+p 2+4.14Proof:Let'=0@uv1A2Xthenwehave:G';'X=u)]TJ/F24 10.909 Tf 10.909 0 Td[(v;u1+Au+)]TJ/F24 10.909 Tf 10.909 0 Td[(v)]TJ/F24 10.909 Tf 10.909 0 Td[(u;v=kuk21)]TJ/F15 10.909 Tf 10.909 0 Td[(Bu;Bv+Au;v+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2)]TJ/F24 10.909 Tf 8.485 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(u;v=kuk21+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(u;vkuk21+)]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2)]TJ/F24 10.909 Tf 10.909 0 Td[(kukkvk:43

PAGE 50

ThenG';'X)]TJ/F24 10.909 Tf 10.909 0 Td[(k'k2X)]TJ/F25 7.97 Tf 12.104 4.295 Td[( 2kvk2)]TJ/F24 10.909 Tf 10.909 0 Td[(kuk21+ 2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(kvk2)]TJ/F24 10.909 Tf 10.909 0 Td[(kuk1kvk2q )]TJ/F24 10.909 Tf 10.909 0 Td[( 2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(kuk1kvk)]TJ/F24 10.909 Tf 18.788 0 Td[(kuk1kvk:Wecancheckthat4)]TJ/F24 10.909 Tf 10.909 0 Td[( 2)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(=22,sothatG';'X)]TJ/F24 10.909 Tf 10.909 0 Td[(k'k2X)]TJ/F24 10.909 Tf 12.105 7.38 Td[( 2kvk20:Theproofiscompleted.Wealreadyestablishedinlemma3.1.1theexistenceoflocalsolutionsforthesystem.11.Nowwewillshowthatthesolutionexistsglobally,whichisadirectconsequenceoftheboundedness.Lemma3.1.3Assumethatthenonlinearityfveries.5,thenanysolution'tofsystem.11existsgloballyforallt0andsatisesk'k2XM2=2 kgk2;fortT1.15forsomeconstantsandT1=T1R;;gwherek'0kR.Proof:Let't=0@utvt1A2Xbeanysolutionofsystem.11withvt=ut+_ut.Takingtheinnerproductof.11with'tinX,weget1 2d dtk'k2X+G';'+fu;_u+fu;u=g;v:.16By.5wehavefu;_u=Xi2Zfui_ui=d dtXi2ZFui;.17fu;u=Xi2ZfuiuiXi2ZFui:.1844

PAGE 51

Sinceg;v1 2kgk2+ 2kvk2,itfollowsfrom.13that1 2d dtk'k2X+2k'k2X+ 2kvk2+d dtXi2ZFui+Xi2ZFui1 2kgk2+ 2kvk2;thatisd dt"k'k2X+2Xi2ZFui#+2k'k2X+2Xi2ZFui1 kgk2:.19Andtaking=inff2;g,wegetd dt"k'k2X+2Xi2ZFui#+"k'k2X+2Xi2ZFui#1 kgk2:.20UsingGronwall'sinequalitywehavek'k2X+2Xi2ZFuie)]TJ/F25 7.97 Tf 6.587 0 Td[(t"k'k2X+2Xi2ZFui;0#+1 kgk2)]TJ/F24 10.909 Tf 10.909 0 Td[(e)]TJ/F25 7.97 Tf 6.586 0 Td[(twhichimpliesk'k2Xe)]TJ/F25 7.97 Tf 6.587 0 Td[(tk'k2X+2 maxku0ksku0kjf0sjku0k2+1 kgk2)]TJ/F24 10.909 Tf 10.909 0 Td[(e)]TJ/F25 7.97 Tf 6.586 0 Td[(t:.21Thisyieldslimt!T'0k'kX<1,sothatthesolution'texistsgloballyforallt>0.NowletR>0,k'0kXRandCR=max)]TJ/F25 7.97 Tf 6.586 0 Td[(RsRjf0sjthenj'k2Xe)]TJ/F25 7.97 Tf 6.587 0 Td[(tR2+CRR2 +1 kgk2:.22Thus.15followswithT1=1 lnR2+CRR2 kgk2andtheproofiscomplete.ThepreviousLemma3.1.3impliesthatequation.11generatesacontinuoussemiowfStgt0onXwhichpossesaboundedabsorbingsetO=fw2X:kwkXMg:.2345

PAGE 52

Thatis:foreveryboundedsetBX,thereaconstantTB>0suchthatStBO;tTB:.24InparticularthereexistsaconstantT0dependingonlyon;gandOsuchthatStOO;tT0:.253.2GlobalAttractorNowthatwehaveestablishedtheexistenceofabsorbingset,itonlyremainstoprovethatthesemiowStisasymptoticallycompacttoconcludetheexistenceofglobalattractor.Firstwepresentsometypeoftailestimate"whichwillbeusefultowardprovingtheasymptoticcompactness.Lemma3.2.1Let'=0@u0v01A2O,thenforevery">0thereexistpositiveconstantsT"andK"suchthatthesolution't=0@utvt1A2Xofsystem.11satisesXjijK"jButij2+jvitj2";tT":.26Proof:Chooseasmoothfunction2C1R+;Rsuchthat0s1fors2R+,ands=0for0s1;s=1fors2:ThenthereexistsaconstantC>0suchthatj0sjCfors2R+.Letkbeaxedpositiveinteger.Setwi=jij kui;zi=jij kui;y=0@wz1A2X.Takeinnerproductof.11withyinXtoget_';yX+G';yX=R';yX:.2746

PAGE 53

Wecancheckthat_';yX=1 2d dtXi2Zjij kj'ij2X;.28wherej'ij2X=jBuij2+jvij2=jui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(uij2+jvij2:.29NowG';y=Bu;Bw)]TJ/F15 10.909 Tf 10.909 0 Td[(Bv;Bw+Bu;Bz+)]TJ/F24 10.909 Tf 10.909 0 Td[(v)]TJ/F24 10.909 Tf 10.909 0 Td[(u;z:.30Let'sestimatethetermsin3.30onebyone.Bu;Bw=Xi2Znhji+1j k)]TJ/F24 10.909 Tf 10.909 0 Td[(jij kiui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(uiui+1+jij kui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(ui2o)]TJ/F15 10.909 Tf 28.128 7.38 Td[(4C0r20 k+Xi2Zjij kui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(ui2;8tT0;Bv;Bw=Xi2Zhji+1j kvi+1)]TJ/F24 10.909 Tf 10.909 0 Td[(viui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(jij kvi+1)]TJ/F24 10.909 Tf 10.909 0 Td[(viuii;Bu;Bz=Xi2Zhji+1j kui+1)]TJ/F24 10.909 Tf 10.91 0 Td[(uivi+1)]TJ/F24 10.909 Tf 10.909 0 Td[(jij kui+1)]TJ/F24 10.909 Tf 10.909 0 Td[(uivii;Bu;Bz)]TJ/F15 10.909 Tf 10.909 0 Td[(Bv;Bw=hji+1j k)]TJ/F24 10.909 Tf 10.909 0 Td[(jij kiui+1vi)]TJ/F24 10.909 Tf 10.909 0 Td[(uivi+1)]TJ/F29 10.909 Tf 28.751 10.364 Td[(Xi2Zj0ij kjui+1vi)]TJ/F24 10.909 Tf 10.909 0 Td[(uivi+1j)]TJ/F15 10.909 Tf 28.128 7.38 Td[(4C0r20 k;8tT0;)]TJ/F24 10.909 Tf 10.91 0 Td[(v)]TJ/F24 10.909 Tf 10.909 0 Td[(u;z=)]TJ/F24 10.909 Tf 10.909 0 Td[(Xi2Zjij kv2i)]TJ/F24 10.909 Tf 10.909 0 Td[()]TJ/F24 10.909 Tf 10.909 0 Td[(Xi2Zjij kuivi)]TJ/F24 10.909 Tf 10.909 0 Td[(Xi2Zjij kv2i)]TJ/F24 10.909 Tf 10.909 0 Td[(Xi2Zjij kuivi:Thus,since<1,wegetthatG';y)]TJ/F15 10.909 Tf 28.128 7.38 Td[(8C0r20 k+Xi2Zjij kjBuij2+)]TJ/F24 10.909 Tf 10.909 0 Td[(Xi2Zjij kv2i47

PAGE 54

)]TJ/F24 10.909 Tf 8.485 0 Td[(Xi2Zjij kuivi8t0:Andfollowingthesameargumentsasintheproofof3.13wecangetG';y)]TJ/F15 10.909 Tf 21.196 7.38 Td[(8C0r20 k+Xi2Zjij khj'ij2X+ 2jvij2i;8tT0:.31Nowweestimatetheright-handsideof.28:R';yX=)]TJ/F15 10.909 Tf 8.485 0 Td[(fu;z+g;z;fu;z=Xi2Zjij kfui_ui+Xi2Zjij kfuiuid dtXi2Zjij kFui+Xi2Zjij kGui;.32g;z=Xi2Zjij kgivi=Xjijkjij kgivi 2Xjijkjij kv2i+1 2Xjijkg2i:g;z 2Xi2Zjij kv2i+1 2Xjijkg2i:.33Substitutinginequalities.28,.31-.33into.27,weobtaind dtXi2Zjij k[j'ij2X+2Fui]+Xi2Zjij k[2j'ij2X+2Fui]8C0r20 k+1 Xjijkg2i:Sinceg2`2,forevery">0,thereexistsaconstantK">0suchthat8C0r20 k+1 Xjijkg2i":ThenfortT0;kK",wehaved dtXi2Zjij k[j'ij2X+2Fui]+Xi2Zjij k[j'ij2X+2Fui]";48

PAGE 55

where=inff2;g.ByGronwall'sinequality,Xi2Zjij k[j'ij2X+2Fui]e)]TJ/F25 7.97 Tf 6.586 0 Td[(t)]TJ/F25 7.97 Tf 6.586 0 Td[(T0Xi2Zjij k[j'iT0j2X+2FuiT0]+" e)]TJ/F25 7.97 Tf 6.586 0 Td[(t)]TJ/F25 7.97 Tf 6.586 0 Td[(T0r201+2 M0+" ;8tT0:whereM0=max)]TJ/F25 7.97 Tf 6.586 0 Td[(r0=sr0=jf0sj.TakingT"=maxnT0;T0+1 ln "+2 M0r20o;thenfortT"andkK"wehaveXjijkj'ij2XXi2Zjij kj'ij2X2" ;.34whichimpliesLemma3.1.Theproofiscompleted.Lemma3.2.2ThesemigroupfStgt0isasymptoticallycompactinX,namely,iff'ngnisboundedinXandtn!1thenfStn'ngnisprecompactinX.Proof:Assumethatk'nkXr;n1forsomepositiveconstantr.By.24thereexistsT,suchthatStf'ngO;8tT.35whereOistheabsorbingsetin.23.Nowsincetn!+1,thereexistsN1rsuchthattnTifnNrwhichimpliesthatStf'ngO;8nN1r.36sothatthereexists'02XandasubsequenceoffStn'ngndenotedstillbyfStn'ngnsuchthatStn'n!'0weaklyinX:.3749

PAGE 56

Wewanttoshowthatthisconvergenceisthestrongsense.Indeedlet">0,byLemma3.2.1and.35thereexistsK1";T">0suchthatXjijK1"kStSTr'nik2X"2 8;tT":Bytn!+1,thereexistsN2r;"suchthattnTr+T"ifnN2r;".Hence,XjijK1"kSTn'nik2X=XjijK1"kStn)]TJ/F24 10.909 Tf 10.909 0 Td[(TrSTr'nik2X"2 8:.38Again,since'02X,thereexistsK2"suchthatXjijK2"k'0k2X"2 8:LetK"=maxfK1";K2"gthenby.37wehaveSTn'nijijK"!'0ijijK"stronglyinR2K"+1asn!+1,thatisthereexistsN3"suchthatXjijK"kSTn'ni)]TJ/F15 10.909 Tf 10.91 0 Td[('0ik2X"2 2;8nN3":.39SettingN"=maxfN1";N2";N3"g,weconcludefornN"thatkSTn'n)]TJ/F24 10.909 Tf 10.909 0 Td[('0k2X=XjijK"kSTn'ni)]TJ/F15 10.909 Tf 10.909 0 Td[('0ik2X+Xjij>K"kSTn'ni)]TJ/F15 10.909 Tf 10.909 0 Td[('0ik2X"2 2+2Xjij>K"kSTn'nik2X)-222(k'0ik2X"2:Theproofiscompleted.50

PAGE 57

Nowwestatethemainresultofthischapterasfollows.Theorem3.2.1Assumethatfsatises.5andg2`2.Then,thedynamicalsystemgeneratedbyequation.11possessesaglobalattractorinX=VHwhichisacompactinvariantsubsetthatattractseveryboundedsetofXwithrespecttothenormtopology.Proof:TheconclusionfollowsfromTheorem1.1.1since,by.24,thereexistsaboundedabsorbingsetandthesemiowisasymptoticallycompactbyLemma3.2.2.51

PAGE 58

4FINALREMARKSWenishthisworkbypresentingsomenalremarksonthedynamicsofevolutionaryequa-tionsinunboundeddomains.Wealsodescribesomeopenproblemsandnewperspectivesinthisarea.FiniteDimensionalityandExponentialAttractorsOnemajorfeatureofglobalattractorsisthattheyusuallyhavenitedimension.Thisreducesthenumberofdegreesoffreedomofthesystemwhichhopefullywillgiveitasimplerdescription.Therearetwoconceptsofdimensionsthataremostlyused:theHausdorandfractaldimensions.However,theglobalattractorhastwomajordrawbacks:ontheonehandtherateofattractioncanbearbitrarilyslowandontheotherhanditisingeneralonlyuppersemicontinuouswithrespecttoperturbationssothattheglobalattractorcanchangeverydrasticallyunderverysmallperturbationsinthestructureoftheoriginaldynamicalsystem.Thisleadstoessentialdicultiesinnumericalsimulationsoftheglobalattractorandevenmakesit,insomesense,unobservable.Inviewofthesedrawbacks,theconceptofexponentialattractorhasbeensuggestedbyEden,Foias,NicolaenkoandTemanin[7].Itisacompact,positivelyinvariantsetwithnitefractaldimension,whichattractstheboundedsetsatanexponentialrate.Itisthereforemorerobustthantheglobalattractorbutitisnotunique.WewillintroducetheconceptsoffractalandHausdordimensionsforgeneralsetsinaBanachspaceX.52

PAGE 59

Denition4.0.1LetAbeasubsetofaBanachspaceX,d>0;">0.Thensetd;"A=infnkXi=1rdi:ri"andAk[i=1Brio;.1whereBridenotesaballofradiusriinX.Itcanbeshownthatd;"Aincreasesas"decreases.Denition4.0.2Wedenethed-dimensionalHausdormeasureofAas:d=sup">0d;"A=lim"!0+d;"A:.2AndtheHausdordimesionofAisdenedby:dHA=inffd>0:dA=0g:.3Astrongermeasureofdimensionisfurnishedbythefractaldimension.Denition4.0.3LetAbeasubsetoftheBanachspaceX.LetN"A=theminimumnumberofballsofradii"thatarenecessarytocoverA.ThenthefractaldimensionofA,dFAisdenedby:dFA=limsup"!0logN"A log1 ":.4AnothercharacterizationofthefractaldimensionisdFA=inffd>0:d;F=limsup"!0"dN"A=0g:.5Next,wegivethedenitionofanexponentialattractorforasemiowfStgdenedonaBanachspaceX.Denition4.0.4AcompactsetMiscalledanexponentialattractororinertialsetforthesemiowfStgt0onXifiSMM,53

PAGE 60

iiMhasnitefractaldimension,dFM,iiitherearepositiveconstantsc0andc1suchthathSnB;Mc0e)]TJ/F25 7.97 Tf 6.587 0 Td[(c1t;8n1:.6Hereh;istheHausdorpseudometricdenedin.5.ForevolutionaryPDEsinanunboundeddomain,therehavebeenresultsonthenitedi-mensionalityoftheglobalattractoraswellastheexistenceofanexponentialattractor,seeforinstance[2],[9],[25].However,therehavebeencounterexamplesoninnitedimensional-ityoftheglobalattractorsee[2],[14],[54],thisimpliesautomaticallythenonexistenceofanexponentialattractor.ThatiswhytheconceptofKolmogorov's"-entropyisexploitedtoobtainsomequalitativeandquantitativeinformationonsuchinnitedimensionalattractors.Itisdenedasfollows.Denition4.0.5LetKbeaprecompactsetinametricspaceMand">0.LetN"K;Mbetheminimalnumberof"-ballsthatcoverK.ThentheKolomogorov's"-entropyofKinMisthefollowingnumber:H"K;M:=lnN"K;M:.7Itisprovedin[15]and[53,54]thatforalargeclassofequationsofmathematicalphysicsinunboundeddomains,the"-entropyoftherestrictionsAjBRx0:=fu0jBRx0;u02AgofthecorrespondingglobalattractortoboundedsubdomainsCRx0,whereCRx0:=x0+[)]TJ/F25 7.97 Tf 10.788 4.295 Td[(R 2;R 2]NistheR-cubeofRNcenteredatx0,satisesH"AjCRx0;L1CRx0CvolCR+Kln+R0="x0ln+R0 ";.8whereln+z:=maxf0;lnzgandtheconstantsC,K,andR0areindependentof",Randx0.ThistypeofestimatesledtotheintroductionofinnitedimensionalexponentialattractorsbyEendiev,MiranvilleandZelikin[10],bymodifyingtheclassicaldenition54

PAGE 61

ofexponentialattractorsandreplacingtheconditionofnitefractaldimensionalitybythe"-entropyestimates.8.Theyprovein[10]thatforcertainreaction-diusionequationsthecorrespondingsemiowadmitsaninnitedimensionalexponentialattractor.Itwouldbeinteresttingtoinvestigatetheexistenceofinnitedimensionalexponentialattractorsforwaveequationsinunboundeddomains.WaveEquationsinExteriorDomainsInthisworkwehaveconsideredwaveequationsinunboundeddomains.Aninterestingclassofsuchequationsarethewaveequationsinanexteriordomain,8>>><>>>:utt)]TJ/F15 10.909 Tf 10.909 0 Td[(u+x;ut=fuin[0;+1ux;0=u0x;utx;0=u1x;anduj@=0;.9whereisanexteriordomain,thatis=RN)]TJ/F24 10.909 Tf 11.444 0 Td[(KforacompactconnectedsubsetK.ThistypeofequationshavebeenextensivelystudiedbyM.Nakao[32,33,34,35,36,37].In[35],heshowedthetotalenergydecayforthecorrespondinglinearequationwhichisappliedtoobtaintheglobalexistenceofniteenergysolutionsforthenonlinearequation.Furthermorehederivedtheenergydecayofthenonlinearequationin[32].Someotherauthorshavealsostudiedsimilarequationsforinstance,Tebou[46]andE.Zuazua[57]amongothers.Thereareinterestingquestionsrelatedtoequation.9.Theenergydecayestablishedin[32],[34],[36]impliesthedissipativityofthesystem.Thequestioniswhetherthereexistsaglobalattractorforsuchsystemsornot?SomeotheropenproblemshavebeenmentionedbyM.Nakaoin[36],forinstance,toderivesomedecayrateoflocalenergyforsolutionsof.9intheparticularcasewherex;t=axjutjrutorx;t=axut+jutjrutwithaxalocalizedfunctiononsomepart!oftheboundary@.55

PAGE 62

OtherPerspectivesTailestimationThetailestimationmethodhasbeencrucialthroughoutourworktoprovetheasymp-toticalcompactnessinthecaseofwaveequationsinunboundeddomainsorforthelatticesystems.In[50],B.Wangprovedthatthistailestimationorasymptoticalnullnessalongwiththeexistenceofaboundedabsorbingsetarenecessaryandsuf-cientfortheexistenceofaglobalattractorforlatticesystems.ItwouldofmuchintereststoinvestigatesuchafeatureforevolutionaryPDEsinunboundeddomains.WaveequationwithoutmasstermWeobtainedtheexistenceofaglobalattractorforthewaveequationwithoutmassterm2.65onlyfordomainsthatareboundedinonedirection.Whathappensforgeneralunboundeddomainsor=RNisstillanopenquestion.56

PAGE 63

References[1]R.A.Adams,Sobolevspaces,2dEd.,AcademicPress,2003.[2]A.V.Babin,M.I.Vishik,Attractorsofpartialdierentialevolutionequationsinanunboundeddomain,Proc.R.Soc.Edinburgh116A1990,221-243.[3]A.Babin,B.Nicolaenko,Exponentialattractorsofreaction-diusionsystemsinanunboundeddomain,J.ofdyn.anddi.Eq.,Vol.7,No.4,,567-590.[4]P.Bates,K.Lu&B.Wang,Attractorsforlatticedynamicalsystems,Int.J.Bifurc.Chaos,11,No.1,,143-153.[5]S.-N.Chow,J.Mallet-Paret,W.Shen,Travellingwaveinlatticedynamicalsystems,J.Di.Eq.149,,248-291.[6]S.-N.Chow,R.Conti,R.Johnson,J.Mallet-Paret,R.Nussbaum,Dynamicalsystems,LectureNotesinMathematics1822,SpringerVerlag,BerlinHaidelberg2003.[7]A.Eden,C.Foias,B.Nicolaenko,R.Temam,Exponentialattractorsfordissipativeevolutionequations,ResearchinAppliedMathematics,vol.37,JohnWiley-Masson,NewYork,1994.[8]A.Eden,V.K.Kalantarov,Onthediscretesqueezingpropertyforsemilinearwaveequations,Tr.J.OfMath.22,335-341.[9]M.Eendiev,A.Miranville,S.Zelik,Exponentialattractorsforanonlinearreaction-diusionsysteminR3,C.R.Acad.Paris,t.330,SerieI,713-718,2000.[10]M.Eendiev,A.Miranville,S.Zelik,Innite-dimensionalexponentialattractorsfornonlinearreaction-diusionsystemsinunboundeddomainsandtheirapproximation,ProceedingsAoftheRoyalSociety460,1107-1129,2004.[11]M.Eendiev,A.Miranville,S.Zelik,Globalandexponentialattractorsfornonlinearreaction-diusionsystemsinunboundeddomains,ToappearinProc.R.Soc.Edin-burgh:SectionA.[12]M.Eendiev,A.Miranville,S.Zelik,Exponentialattractorsandnite-dimensionalreductionfornonautonomousdynamicalsystems,ToappearinProc.R.Soc.Edinburgh:SectionA.57

PAGE 64

[13]M.Eendiev,A.Miranville,S.Zelik,ExponentialattractorsforasingularlyperturbedCahn-Hillardsystem,ToappearinMathematischeNachrichten.[14]M.Eendiev,S.Zelik,TheAttractorforaNonlinearReaction-DiusionSystemintheUnboundedDomain,Comm.PureAppl.Math.54,No.6,625-688.[15]M.Eendiev,S.Zelik,UpperandlowerBoundsfortheKolmogorovEntropyoftheAttractorforaReaction-DiusionEquationinanUnboundedDomain,J.Dyn.Di.Eqns.14,No.2,369-403.[16]E.Feireisl,AttractorsforsemilineardampedwaveequationsonR3,NonlinearAnalysis23,N0.2,,187-195.[17]E.Feireisl,BoundedLocallycompactglobalattractorsforsemilineardampedwaveequa-tionsonRN,J.Di.andInt.Equations9,No.5,1147-1156.[18]E.Feireisl,Long-timebehaviorandconvergenceforsemilinearwaveequationsinRN,J.Dyn.ofDiEq.9,No.11997,133-155.[19]E.Feireisl,Onthelong-timebehaviourofsolutionstononlineardiusionequationsonRn,Nonl.Diif.eq.appl.4,43-60.[20]E.Feireisl,P.Laurencot,F.Simondon,H.Toure,Compactattractorsforreaction-diusionequationsinRN,C.R.Acad.Sc.Paris,t.319,SerieI,,147-151.[21]V.Georgiev,S.Lucente,WeightedSobolevspacesappliedtononlinearKlein-Gordonequation,C.R.Acad.Sc.Paris,t.329,SerieI,,21-26.[22]M.Gobbino,TopologicalPropertiesofattractorsfordynamicalsystems,Topology40,,279-298.[23]J.KHale,AsymptoticBehaviorofDissipativeSystems,MathematicalSurveysandMonographs,Vol25,AMS,Providence1988.[24]N.I.Karachalios,N.M.Stavrakakis,ExistenceofaglobalattractorforsemilineardissipativewaveequationsonRN,J.Di.Eq.157,,183-205.[25]N.I.Karachalios,N.M.Stavrakakis,EstimatesonthedimensionofaglobalattractorforasemilineardissipativewaveequationonRN,Disc.Cont.Dyn.Sys.Vol.bf8,No.4,,939-951.[26]A.Kufner,WeightedSobolevsapces,JohnWiley&Sons1985.[27]W.Liu,B.Wang,DynamicsoftheFitzHugh-Nagumosystem,2004,Preprint.[28]K.Lu,B.Wang,GlobalAttractorsfortheKlein-Gordon-SchrodingerEquationinUn-boundedDomains,J.ofDierentialEquations170,,281-316.[29]A.Lunardi,AnalyticSemigroupsandOptimalRegularityinParabolicProblems,Birkhausser,Basel,1995.58

PAGE 65

[30]A.Miranville,S.Zelik,Uniformexponentialattractorsforasingularlyperturbeddampedwaveequation,Disc.Cont.Dyn.Sys.Vol.bf9,No.4,2003,1-27.[31]S.Merino,OntheexistenceofthecompactglobalattractorforsemilinearreactiondiusionsystemsonRN,J.Di.Eq.132,,87-106.[32]M.Nakao,Decayofsolutionsofthewaveequationwithsomelocalizeddissipations,Nonl.An.,TMA,30,No.6,,3775-3784.[33]M.Nakao,Stabilizationoflocalenergyinanexteriordomainforthewaveequationwithalocalizeddissipation,J.Di.Eq.148,,388-406.[34]M.Nakao,EnergyDecayforthelinearandsemilinearwaveequationsinexteriordo-mainswithsomelocalizeddissipations,Math.Z.238,,781-797.[35]M.Nakao,Globalexistenceforsemilinearwaveequationsinexteriordomains,Nonl.An.47,,2497-2506.[36]M.Nakao,DecayandGlobalExistenceforNonlinearWaveEquationswithlocalizeddissipationsinGeneralExteriorDoamins,2004,Monographpreprint.[37]J.J.Bae&M.Nakao,ExistencefortheKirchhotypewaveequationwithalocalizedweaklynonlineardissipationinexteriordomains,Disc.Cont.Dyn.Sys.11,Nos.2&3,,731-743.[38]A.Pazy,Semigroupsoflinearoperatorsandapplicationstopartialdierentialequa-tions,AppliedMathematicalSciences,vol.44,SpringerVerlag,NewYork1983.[39]K.Puger,NonlinearboundaryvalueproblemsinweightedSobolevspaces,Nonl.An.TMA,30.No.2,,1263-1270.[40]M.Prizzi,Aremarkonreaction-diusionequationsinunboundeddomains,Disc.Cont.Dyn.syst.,9,No.2,,281-286.[41]R.Rosa,Theglobalattractorforthe2DNavier-Stokesowonsomeunboundeddo-mains,Nonl.An.TMA,32,No.1,,71-85.[42]G.R.Sell,Y.You,DynamicsofEvolutionaryEquations,Springer,NewYork,2002.[43]M.Stanislavova,A.Stefanov,B.Wang,AsymtoticsmoothingandattractorsforthegeneralizedBenjamin-Bona-MahonyequationonR3,2004Preprint.[44]W.AStrauss,Nonlinearwaveequations,Conferenceboardformathematicalsciences,Amer.Math.Soc.,73,1989.[45]M.Struwe,SemilnearWaveEquations,Bull.Amer.Math.Soc.26,N0.1,,53-85.[46]L.R.T.TebouOnthedecayestimatesforthewaveequationwithalocaldegenrateornondegeneratedissipation,PortugaliaeMathematica,Vol.55Fasc.3.59

PAGE 66

[47]R.Temam,Innite-DimensionalDynamicalSystemsinMechanicsandPhysics,Springer,Berlin,1988.[48]E.V.Vleck,B.Wang,LatticeFitzHugh-Nagumosystems,2004,Preprint.[49]B.Wang,Attractorsforreaction-diusionequationinunboundeddomains,PhysicaD128,41-52.[50]B.Wang,Dynamicsofsystemsofinnitelattices,2004,Preprint.[51]Y.You,Globaldynamicsofnonlinearwaveequationswithcubicnon-monotonedamp-ing,DynamicsofPDE,Vol.1,No.1,65-86.[52]Y.You,Spectralbarriersandinertialmanifoldsfortime-discretizeddissipativeequa-tions,Comp.&Math.withAppl.,48,,1351-1368.[53]S.V.Zelik,Theattractorforanonlinearhyperbolicequationintheunboundeddomain,Disc.Cont.Dyn.Sys.,7,No.3,,593-611.[54]S.Zelik,TheAttractorforaNonlinearReaction-DiusionSysteminanUnboundedDomainandKolmogorov'sEpsilon-Entropy,Math.Nachr.232,No.1,129-179.[55]S.Zheng,Nonlinearevolutionequations,MonographsandsurveysinPureandAppliedMathematics133,Chapman&Hall/CRC,2004.[56]S.Zhou,Attractorsforsecondorderlatticedynamicalsystems,J.DierentialEquations179,605-624.[57]E.Zuazua,Exponentialdecayforthesemilinearwaveequationwithlocalizeddampinginunboundeddomains,J.Math.Puresetappl.,70,,513-529.60

PAGE 67

AbouttheAuthorDjibyFallwasbornin1974inLinguere,Senegal.HereceivedhisBachelorandMastersinAppliedMathematicsattheUniversityGastonBergerinSaint-Louis,Senegal.Intheyear2000-2001heparticipatedintheDiplomaprogrammeinMathematicsattheAbdusSalamInternationalCentreforTheoreticalPhysicsICTP,inTrieste,Italy.IntheFall2001hewasadmittedintothegraduateprograminmathematicsattheUniversityofSouthFloridaUSF,inTampawherehestartedworkingunderthesupervisionofProfessorYunchengYou.HisscholarlyinterestsareinPDEs,DynamicalSystems,mathematicalmodelinginnancialeconomicsanduiddynamics.