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1、DimensionofattractorsofnonautonomouspartialdifferentialequationsT.Caraballo¤J.A.LangayJ.ValerozAbstractTheconceptofnonautonomous(orcocycle)attractorhasbecomeapropertoolforthestudyoftheasymptoticbehaviourofgeneralnonau-tonomouspartialdifferentialequations.Thisisatime-dependentf
2、am-ilyofcompactsets,invariantfortheassociatedprocessandattracting“from¡1”:Ingeneral,theconceptisratherdifferentfromtheclas-sicaloneofglobalattractorforautonomousdynamicalsystems.Weproveageneralresultonthefinitefractaldimensionalityofeachcom-pactsetofthisfamily.Inthisway,wegene
3、ralizepreviousresultsofChepyzhovandVishikin[6].Ourresultsarealsoappliedtodiffer-entialequationswithanonlineartermhavingpolynomialgrowthatmost.Contents1Introduction22Attractorsofnonautonomousequations33Dimensionofnonautonomousattractors54Applicationstoanonautonomouspartialdiffe
4、rentialequa-tion10¤Dpto.EcuacionesDiferencialesyAn´alisisNum´erico,UniversidaddeSevilla,Apdo.deCorreos1160,41080-Sevilla,Spain.E-mail:caraball@us.esyDpto.EcuacionesDiferencialesyAn´alisisNum´erico,UniversidaddeSevilla,Apdo.deCorreos1160,41080-Sevilla,Spain.E-mail:langa@numer
5、.us.eszUniversidadCardenalHerreraCEU,Comisario3,03203Elche,Alicante,Spain.E-mail:valer.el@ceu.es11IntroductionInthispaper,wedevelopageneraltheoryonthefinitedimensionofat-tractorsfornonautonomouspartialdifferentialequationsandweapplyit,inparticular,toestimatethefractaldimension
6、oftheattractorforthefollowingnonautonomousequation8><@u¡∆u+f(t;u)=h(t);@t>:uj@Ω=0;u(¿)=u¿;wherethefunctionh(t)isallowedtohavepolynomialgrowthintime(seecondition(9)below).Forthesekindofnonautonomoussystemsitisnotpos-sibleingeneraltoobtainauniformglobalattractorinthesenseof[5]
7、,sincethetrajectoriescanbeunboundedwhentimerisestoinfinity.Adifferentapproachwasdevelopedin[8],[9],[25](seealso[2],[17],[16],[24]),wheretheexistenceofattractorsforsomestochasticandnonautonomousequationsisstudied.Themaindefinitionsandtheoremsfromtheabstracttheoryofattractorsfors
8、uchsystemsaregiveninSection2.Itisworthpointingoutthatinsuchsystemstheglobal
