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1、DISCRETEANDCONTINUOUSDYNAMICALSYSTEMSVolume6,Number4,October2000pp.875–892STABILITYANDRANDOMATTRACTORSFORAREACTION-DIFFUSIONEQUATIONWITHMULTIPLICATIVENOISETomasCaraballo&Jos´eA.Langa´Dpto.EcuacionesDiferencialesyAn´alisisNum´erico,UniversidaddeSevilla,Apdo.deCorreos1160,41080
2、-Sevilla.Spain.JamesC.RobinsonMathematicsInstitute,UniversityofWarwick,CoventryCV47AL.U.K.Abstract.Westudytheasymptoticbehaviourofareaction-diffusionequation,andprovethattheadditionofmultiplicativewhitenoise(inthesenseofItˆo)stabilizesthestationarysolutionx≡0.Weshowinadditiont
3、hatthisstochasticequationhasafinite-dimensionalrandomattractor,andfromourresultsconjectureapossiblebifurcationscenario.1.Introduction.Thestudyoftheasymptoticbehaviourofevolutionequationsisoneofthemostimportantproblemsinthefieldofdifferentialequations,asthevastliteratureonthesubj
4、ectshows.Perhapsthefirststepinthisprogrammeistoinvestigatethestabilityofthestationarysolutions,andthisisatpresentawell-developedbranchofthetheoryofstochasticordinaryandpartialdifferentialequations(see,amongothers,Has’minskii[20],Mao[23],[24],CaraballoandLiu[5]).Suchastabilityan
5、alysisisessentiallyalocalstudy,asweobtaininformationonthedynamicsonlyaroundthesepoints.Amorecompleteanalysisofthequali-tativepropertiesofthesystemasawholerequiresinformationrelatingnotonlytothestationarypointsbutalso,forexample,tothestabilityorinstabilityoftheas-sociatedinvar
6、iantmanifoldsofthesepoints.Asafurthersteponecanconsidertheglobalattractorfortheproblem(Ladyzhenskaya[22],Hale[19],Temam[32]);thisisrapidlybecomingoneofthemainconceptsinthetheoryofinfinite-dimensionaldynamicalsystems.Recently,CrauelandFlandoli[9](seealsoSchmalfuß[30])havegenera
7、lizedthetheoryofdeterministicattractorstothestochasticcase.Thistheoryofrandomattractorsisturningouttobeveryfruitfulinthestudyofthelong-timedynamicsofstochasticordinaryandpartialdifferentialequations.Inthispaperwestudyareaction-diffusionequationperturbedbyamultiplicativewhitenoi
8、seterm,firstconsideringthisintheItˆosense,dx(t)=∆x(t)dt+(ax(t)−x(t)3)
