cahn-川东-西藏中国-美国-印度-美国中国

cahn-川东-西藏中国-美国-印度-美国中国

这篇文章与学习报告中的短程风险相比。从这一点可以看出，有一个短程投诉被纳入投诉。calendary分布模式，u与非本有明确的投诉分为以下投诉。{du+(vΔ2u-Δf(u))dt=bu˚dW(t)(x‚t)∈G×Ru=0onτ(1)Wherev＞0‚b＞0‚G=n∏i=1]0‚Li[‚Li＞0,theperiodicboundaryconditionisϕ|xi=0=ϕ|xi=Lii=1‚2‚⋯‚n(2)Theorderforuandthederivativesofuisatleastof3.fisapolynomialoforder2p-1,andisdefindbyf(u)=2p-1∑j=1ajujp∈Ν‚p≥2‚a2p-1＞0(3)Also,wesupplementequation(1)withaninitialconditionu(0‚x)=u0(x)(4)ThewhitenoisedescribedbyaprocessW(t)resultfromthefactthatsmallregularityhastobetakenaccountinsomecircumstances.Here,weassumethat,W(t)isatwo-sidedWienerprocessontheprobabilityspace(Ω,F,P),whereΩ={ω∈C(R,R):ω(0)=0},FistheBorelsigma-algebrainducedbythecompact-opentopologyofΩ,PisaWienermeasure.Ofcourse,theuniquesolutionofequations(1)-(4)willgeneratearandomdynamicalsystem(RDS),thisallowsustoconsidertherandomattractoroftheRDS.1Let(Ω,F,P)beaprobabilityspace,and{θt:Ω→Ω:t∈R}afamilyofmeasurepreservingtransformationssuchthat(t‚ω)|→θtωismeasurable,θ0=id,andθt+s=θtθsforalls,t∈R.Theflowθttogetherwiththecorrespondingprobabilityspace(Ω,F,P,θt)iscalledameasurabledynamicalsystem.Acontinuousrandomdynamicalsystem(RDS)onaPolishspace(X,d)withBorelsigma-algebraBoverθon(Ω,F,P)isameasurablemapφ∶suchthat(ⅰ)φ(0,ω)=idonX;(ⅱ)φ(t+s,ω)=φ(t,θsω)φ(s,ω)foralls,t∈R+(cocycleproperty);(ⅲ)φ(t,ω):X→Xiscontinuous.Arandomcompactset{K(ω)}ω∈Ωisafamilyofcompactsetsindexedbyωsuchthatforeveryx∈X,themapω|→d(x‚Κ(ω))ismeasurablewithrespecttoF.LetA(ω)bearandomsetandB⊂X.WesayA(ω)attractsBiflimt→∞dist(φ(t‚θ-tω)B‚A(ω))=0Wheredist(·,·)denotestheHausdorffsemi-distanceinX.WesayA(ω)absorbsB,ifthereexiststB(ω)suchthatφ(t,θ-tω)B⊂A(ω)forallt≥tB(ω).ArandomsetA(ω)issaidtobearandomattractorfortheRDSφif(ⅰ)A(ω)isarandomcompactset;(ⅱ)A(ω)isinvariant,thatis,φ(t,ω)A(ω)=A(θtω)forallt≥0;(ⅲ)A(ω)attractsalldeterministicboundedsetsB⊂X.Theorem1IfthereexistsarandomcompactsetabsorbingeveryboundedsetB⊂X,thentheRDSφpossessesarandomattractorA(ω)=∪B⊂XΛB(ω)¯whereΛB(ω)=∩s≥0∪t≥sφ(t‚θ-tω)B¯istheomega-limitsetofB.2u2004unequess.3.3s/asis-nr2which/which病料可达市政府分类s.........................................................3.3.3.Lettheprocessα(t)=e-bW(t)thendα=-bα。dW.Ifweletξ=αu,wegetthenewequation(nostochasticdifferentialappearshere){dξdt+vΔ2ξ+αΔf(u)=0ξ=0onτ(5)Tosolvetheequation,weconsidertheL2(G)spacewiththescalarproduct(·,·)andnorm|·|,wedenoteby|G|theLebesguemeasureofG‚m(u)=1|G|∫Gu(x)dx‚L˜2(G)isasubspaceofL2(G).ThespaceL˜2(G)wasdefinedbyL˜2(G)={u∈L2(G):m(u)=0},letu˜=u-m(u)(∀u∈L2(u)),thenu˜istheprojectionoperatorsinL˜2(G).WedefinethelinearunboundedoperatorAonL˜2(G)byAu=-Δuu∈D(A)=Ηper2(G)TheoperatorAisself-adjointpositiveandhascompactresolvent.ItpossessesabasisofeigenvectorseiwhichisorthonormalinL2(G),itisassociatedwiththeeigenvaluesλi0=λ0＜λ1≤λ2≤⋯≤λj→+∞Thefunctione0isconstantandequalto|G|-12.Obviously,m(u)=1|G|12(u‚e0)u∈L2(G).Usingthisabstractsetting,werewriteequation(5)asddtA-1ξ+vAξ+αf(u)=0(6)ForallsinR,AsisdefinedbyAsu=∑j=1∞λjsujeju=∑j=0∞ujejThedomainofAs2isVs=D(As2)={u=∑j=0∞ujej∶∑j=1∞λjsuj＜+∞}itisendowedwiththeseminormandthesemiscalarproduct|u|s=|As2u|(u‚v)s=(As2u‚As2v)andwiththenorm∥u∥s=(|u|s2+m2(u))12Moreover,V0=L2(Ω),and(u˜‚v˜)=(u˜‚v)=(u‚v)0∀u‚v∈V0Whensbelongsto[0,2],VsisaclosedsubspaceofHs(G),and‖·‖sisequivalenttotheusualnormdefined.ItisclearthatthefollowingtypePoincareinequality{|u|s≤λ1s-s´|u|s´s≤s´‚u∈Vs´∥u∥-1≤c1´|u|0∀u∈V0|u|0≤c2´∥u∥1∀u∈V1(7)andtheinterpolatoryinequality|u|λs1+(1-λ)s2≤|u|s1λ|u|s21-λs1≤s2‚λ∈[0‚1]‚u∈Vs2(8)hold.UsingtheGalerkinapproximationandaprioriestimates,similarlytotheproofofTheorem3.3.1ofreference,onecanshowthat(ⅰ)Forallt0∈Randu0∈V-1,thereexistsauniquesolutionu∈C([t0,∞);V-1)ofequation(5)withu(t0)=u0;(ⅱ)Ifu0∈L2(G),thesolutionu∈L∞(0,T;L2(G));(ⅲ)Foreveryε>0,u∈C([t0+ε,∞);V-1)foru0∈V-1;(ⅳ)Denotingsuchsolutionbyu(t,ω;t0,u0),themappingu0|→u(t‚ω;t0,u0)iscontinuousforallt≥t0.Byabove(ⅰ)-(ⅳ),itiseasytoprovethat,φisacontinuousRDSonV-1overθon(Ω,F,P),whereθisameasurabledynamicalsystemdeterminedbythetwo-sidedWienerprocessW(t,ω)=ω(t)withθsω(t)=ω(t+s)-ω(s)∀s‚t∈RWealsohavethefollowingfacts:(ⅴ)u=φ(t,ω)u0isjusttheuniquesolutionofequation(5)withu(0)=u0;(ⅵ)φ(s,θ-sω)u0=u(0,ω,-s,u0)foru0∈V-1,andcanbeinterpretedasthepositionattime0ofthetrajectorywhichwasatu0attime-s.3u2004u2004InthisSection,wewillshowthatthereexistsacompactrandomattractorforRDSφasintroducedinSection2.ByTheorem1,wehavetogetacompactrandomabsorbingset.Lemma1Assumeξisasolutionofequation(5),then∥ξ(t)∥-12≤e-2vc0c1t[∥ξ(t0)∥-12e2vc0c1t0+∫t0tc2´α2(s)e2vc0c1sds]∀t≥t0ProofTakingthescalarproductofequation(5)withξ,andusingtheGreenformula,weobtain(ddtA-1ξ‚ξ)+(vAξ‚ξ)+α(f(u)‚ξ)=012ddt∥ξ∥-12+v∥ξ∥12+α(f(u)‚ξ)=012ddt∥ξ∥-12+v∥ξ∥12+α2(f(u)‚u)=0UsingtheYonginequality,weinferfromformula(3)theexistenceofaconstantc′1>0suchthat∥∑j=1j=2p-2ajuj+1∥-1≤12a2p-1u2p+c1´∀uandhence12a2p-1u2p-c1´≤f(u)u≤32a2p-1u2p+c1´ddt∥ξ∥-12+2v∥ξ∥12+2α2∫Ωf(u)udx=0ddt∥ξ∥-12+2v∥ξ∥12+α2∫Ωa2p-1u2pdx≤2α2c1´|Ω|ddt∥ξ∥-12+2v∥ξ∥12≤c2´α2c2´=2c1´|Ω|(9)where|Ω|isthemeasure(volume)ofΩ.DuetothePoincareinequality,thereexistsconstantsc0,c1,suchthat∥ξ∥-1≤c0c1∥ξ∥1Weinferfrom(9)thatddt∥ξ∥-12+2v1c0c1∥ξ∥-12≤c2´α2UsingtheclassicalGronwalllemma,weseethat∥ξ(t)∥-12≤∥ξ(t0)∥-12e∫t0t-2vc0c1dτ+∫t0tc2´α2(s)e-∫ts-2vc0c1dτds∥ξ(t)∥-12≤e-2vc0c1t[∥ξ(t0)∥-12e2vc0c1t0+∫t0tc2´α2(s)e2vc0c1sds](10)Proposition1Thereexistsarandomvariableγ(ω)satisfyingthefollowingproperty:Foreveryρ>0,thereexistst(ω,ρ)≤-1,suchthatforanyu0∈V-1with‖u0‖-1<ρ,andforanyt0≤t(ω,ρ),thefollowingsestimateshold∥ξ(t‚ω；t0‚α(t0‚ω)u0)∥-1≤γ(ω)∀t∈[-1‚0](11)Inparticular,B(0,γ(ω))isarandomabsorbingsetforRDSφinV-1.ProofToproveestimate(11),weseefromformula(10)that∥ξ(t‚ω；t0‚α(t0‚ω)u0)∥-12≤e2vc0c1[∥u0∥-12α2e2vc0c1t0+∫t0tc2´α2(s)e2vc0c1sds](12)holdsforall-1≤t≤0andt0≤-1.Byastandardargument(seereference),wehavelims→-∞W(s)s=0(13)Iteasilyfollowsthatα2(t0)e2vc0c1t0→0t0→-∞whichimpliesthatthereexistst1(ω,ρ)≤-1,suchthatift0≤t1,and∥u0∥-12α2(t0‚ω)e2vc0c1t0≤ρ2α2(t0‚ω)e2vc0c1t0≤1(14)andthereexistst2(ω,ρ)≤-1,suchthatift0≤t2,and∫t0tc2´α2(s)e2vc0c1sds≤c2´∫-∞0α2(s)e2vc0c1sds≤γ1(ω)Whereγ1(ω)isarandomvariablebeingindependentofρ.Becausetheformula(13)implieseasilythat,thereexistsε,ifε=2vbc0c1subjectto|w(s)s|＜ε2b,thenα2(s)e2vc0c1s≤e-εse2vc0c1s=e(2vc0c1-ε)s,togetherwithformula(13),itimpliesformula(12)bytakingt(ω,ρ)=min{t1,t2},andγ2(ω)=e2vc0c1(1+γ1(ω)).Finally,sinceφ(-t0,θt0ω)ξ0=ξ(0,ω;t0,α(t0,ω)u0),itfollowsfromeastimate(11)that|φ(-t0,θt0ω)ξ0|≤γ(ω),wheret0≤t(ω,ρ),whichmeansthatB(0,γ(ω))isarandomabsorbingsetinV-1.Lemma2UnderthesameasumptionandnotationsofProposition1,thereexistsonerandomvariableC1(ω),suchthatfort0≤t(ω,ρ),and∫-10∥ξ(s‚ω；t0‚α(t0‚ω)u0)∥12ds≤C1(ω)Byintegratingformula(9)between-1and0,wehave∫-10ddt∥ξ∥-12ds+∫-102v∥ξ∥12ds≤∫-10c2´α2ds∥ξ(0)∥-12-∥ξ(-1)∥-12+2v∫-10∥ξ∥12ds≤c2´∫-10α2ds2v∫-10∥ξ∥12ds≤c2´∫-10α2ds+∥ξ(-1)∥-12Byestimate(11),weobtain2v∫-10∥ξ∥12ds≤c2´∫-10α2ds+γ2(ω)Thus∫-10∥ξ∥12ds≤c2´2v∫-10α2(s)ds+12vγ2(ω)=C1(ω)Proposition2ThereexistsonerandomradiusC2satisfyingthefollowingproperties:Forallρ>0,thereexistst(ω,ρ)≤-1,suchthatforallt0≤t(ω,ρ)andu0∈Hwith|u0|-1<ρ,thefollowingshold|ξ(0‚ω；t0‚α(t0‚ω))u0|≤C2(ω)Inparticular,theballB(0,C2(ω))isarandomabsorbingsetinL˜2(Ω).ProofTakingthescalarproductoftheequation(6)withAξinL˜2(G),weobtain(ddtA-1ξ‚Aξ)+(vAξ‚Aξ)+α(f(u