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1、Attractorsfornonlinearreaction-diffusionsystemsinunboundeddomainsviathemethodofshorttrajectoriesMaurizioGrasselliDipartimentodiMatematicaPolitecnicodiMilanoViaBonardi9I-20133Milano,ItalyE-mail:maurizio.grasselli@polimi.itDaliborPraˇz´akKatedramatematick´eanal´yzyMatematicko-fyzik´aln´ıfakultaUni
2、verzityKarlovySokolovsk´a83,CZ-18675,CzechRepublicE-mail:prazak@karlin.mff.cuni.czGiulioSchimpernaDipartimentodiMatematica,Universit`adiPaviaViaFerrata,1,I-27100Pavia,ItalyE-mail:giusch04@unipv.itOctober19,2009AbstractdWeconsideranonlinearreaction-diffusionequationonthewholespaceR.Weprovethewell
3、-posednessofthecorrespondingCauchyprobleminageneralfunctionalsetting,namely,whenthe2initialdatumisuniformlylocallyboundedinL.Thenweadapttheshorttrajectorymethodtoestablishtheexistenceoftheglobalattractorand,ifd≤3,wefindanupperboundofitsKolmogorov’sε-entropy.Keywords:reaction-diffusionsystem,unbou
4、ndeddomain,globalattractor,Kolmogorov’sε-entropy.AMS(MOS)subjectclassification:35B41,35K57,92D25.arXiv:0910.3588v1[math.AP]19Oct20091IntroductionTheasymptoticbehaviorofsolutionstoreaction-diffusionequationshasbeentheobjectofalargenumberofinvestigations.Inparticular,theexistenceofglobalandexponent
5、ialattractorsandtheirfractaldimensionhavebeencarefullystudiedinthecaseofboundeddomains(see,e.g.,[17]andreferencestherein).However,unboundeddomainsarealsoratherinterestingforapplications.Inthiscase,thedynamicscanexhibitamorecomplexbehaviorcharacterized,forinstance,bytravellingwavesconnectingcons
6、tantequilibriaorbyacontinuumofspaceperiodicequilibria(andmuchmore,asshownin[24]).Thelackofstandardcompactinjectionsrequirenewideasandthechoiceofthetopologybecomescrucialinordernottoexcludeinterestinginvariantsolutions.Inthepioneeringpapersontheexistenceofglobalattractors[1,4]weightednormswerein
7、troducedandused.Sincethen,manycontributionshavefollowed(see,e.g.,[2,3,8,9,11,16,18,19,22]andreferencestherein)makinguseofweightedornotphasespacesandundervariousassumptionsonthereaction(and,possibly,convective)terms.However,asnotic
