Random attractos for S-G equation.pdf

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1、PACIFICJOURNALOFMATHEMATICSVol.216,No.1,2004RANDOMATTRACTORFORADAMPEDSINE–GORDONEQUATIONWITHWHITENOISEXiaomingFanWeprovetheexistenceofacompactrandomattractorfortherandomdynamicalsystemgeneratedbyadampedSine–Gordonwithwhitenoise.Andweobtainapreciseestimateoftheupperboundofth

2、eHausdorffdimensionoftherandomattractor,whichdecreasesasthedampinggrowsandshowsthatthedimensionisuniformlyboundedforthedamping.Inparticular,undercertainconditions,thedimensioniszero.1.IntroductionThisarticleisdevotedtotheexistenceandestimateofHausdorffdimensionoftherandomattr

3、actorforadampedSine–Gordonequationwithhomom-geneousDirichletboundaryconditionwhenthereisarandomterm.LetΩbeanopenboundedsetofRnwithasmoothboundary∂Ω.Weconsidertheequationutt+αut−∆u+βsinu=q(x)W,˙inΩ×[τ,+∞),τ∈R,(1)u(x,t)

4、x∈∂Ω=0,t≥τ,u(x,τ)=u0(x)∈H01(Ω),ut(x,τ)=u1(x)∈L2(Ω).

5、whereu=u(x,t)isareal-valuedfunctiononΩ×[τ,+∞),forτ∈R;α>0iscalledthedamping;q(x)∈H2(Ω)∩H01(Ω),W˙(t)isthederivativeofaone-dimensionaltwo-sidedWienerprocessW(t);andq(x)W˙formallydescribeswhitenoise.Arandomattractorofarandomdynamicalsystem(RDS)isameasurableandcompactinvariantra

6、ndomsetattractingalltheorbits.Whensuchanattractingexists,itisthesmallestattractingcompactsetandthelargestinvariantset[3].Thisseemstobeagoodgeneralizationofthenowclassi-calconceptofaglobalattractorfordeterministicdynamicalsystems.Thenotionofarandomattractorisveryusefulforman

7、yinfinite-dimensionalrandomdynamicalsystems[4,3].Manyauthors[8,10]havestudiedandestimatedtheHausdorffdimensionoftheglobalattractorforadeterministicdampedSine–Gordonequation.H.CrauelandF.Flandoli[4]introducedthenotionofarandomattractorandobtainedageneraltheoremontheexistenceof

8、arandomattractorforan6364XIAOMINGFANRDS.Theirtheoremhasbeensuccessfullyappliedtothestochasticreaction-diffusionequationandthestochasticNavier–Stokesequation.H.Craueletal.[3]generalizedthenotionofanattractorforthestochasticdynamicalsystemintroducedpreviouslyandconsideredastoc

9、hasticnonlinearwaveequation.A.Debussche[6,7]providedageneralwaytoobtaintheHausdorff