Sharp Transition Between Extinction and Propagatio

arXiv:math/0504333v1[math.AP]15Apr2005SHARPTRANSITIONBETWEENEXTINCTIONANDPROPAGATIONOFREACTIONANDREJZLATOˇSAbstract.Weconsiderthereaction-diffusionequationTt=Txx+f(T)onRwithT0(x)≡χ[−L,L](x)andf(0)=f(1)=0.In1964Kanel’provedthatiffisanignitionnon-linearity,thenT→0ast→∞whenLL0,andT→1whenLL1.WeanswertheopenquestionofrelationofL0andL1byshowingthatL0=L1.WealsodeterminethelargetimelimitofTinthecriticalcaseL=L0,thusprovidingthephaseportraitfortheabovePDEwithrespecttoa1-parameterfamilyofinitialdata.Analogousresultsforcombustionandbistablenon-linearitiesareprovedaswell.1.IntroductionInthepresentpaperweconsiderthereaction-diffusionequationTt=ΔT+f(T)(1.1)inthecylinderR×ΩwhereΩisadomaininRn−1,withNeumannboundaryconditionsonR×∂Ω.Thenon-linearreactiontermfisassumedtobeLipschitzcontinuouswithf(0)=f(1)=0andtheinitialdatumT0isbetween0and1.WewilltreatthecasewhenT0isindependentofthetransversalvariabley∈Ω,andso(1.1)becomesTt=Txx+f(T)(1.2)withx∈R.Thisequationhasbeenextensivelystudiedinmathemati-cal,physicalandotherliterature,startingwiththepioneeringworksofFisher[7]andKolmogorov,Petrovskii,Piskunov[11].Inthesepapers(1.2)wasusedtodescribethepropagationofadvantageousgenesinapopulation.Themainobjectofstudyintheseandmanysubsequentworkswastheexistenceandstabilityoftravelingfrontsfor(1.2)and(1.1).Intherecentyearsmostoftheresultshavebeenextendedtoincludeanadvectiontermu·∇Tin(1.1),andwerefertothereviews[2,16]foranextensivebibliography.1991MathematicsSubjectClassification.Primary:35K57;Secondary:35K15.12ANDREJZLATOˇSTheaboveequationsareusedtomodelnotonlypopulationgeneticsphenomena.Whenf(θ)0forθ∈(0,1),thenfisacombustionnon-linearityand(1.1)/(1.2)modelanexotermicchemicalreactioninaninfinitetubewithazeroheat-lossboundary,inparticular,flamepropa-gationinapremixedcombustiblegaswithoutadvection(seeZel’dovichandFrank-Kamenetskii[17]).InthissettingTisthenormalizedtem-peraturetakingvaluesin[0,1].Wenotethat(1.1)isusuallyobtainedfromasysteminvolvingboththetemperatureandtheconcentrationofthereactantsafterthesimplifyingassumptionofequalthermalandmaterialdiffusivities.Aspecialcaseofpositivef,usedofteninchemicalandbiologicalliterature,istheKPPtypewithf′′(θ)≤c0[11].Incombustionmodelsthenon-linearityisoftenconsideredtobeofArrheniustypewithslowreactionratesatlowtemperatures,modeledbyf(θ)=e−A/θ(1−θ).Oneoftenapproximatesthissituationbyconsideringanignitionnon-linearityfsatisfyingf(θ)=0forθ∈[0,θ0]andf(θ)0forθ∈(θ0,1),withθ0∈(0,1)theignitiontemperature.Thethirdprominentcaseisthebistablenon-linearitywithf(θ)0forθ∈(0,θ0)andf(θ)0forθ∈(θ0,1),whereoneusuallyassumesR10f(θ)dθ0.Thishasbeenusedtomodelsignalpropagationalongbistabletransmissionlines,inparticular,nervepulsepropagation[12].Inbiologicalcontextitisalsocalledheterozygoteinferior(seeAronsonandWeinberger[1]).Inthispaperwewillconsideralltheabovetypes.Ourinterestherewillnotbeinthequestionoftravelingfronts,butinextinctionofreaction—quenchingofflames.WewillthereforeassumetheinitialdatumT0(x)for(1.2)tobecompactlysupported,andwillwanttoknowwhenkT(t,·)k∞→0ast→∞.(1.3)ForthesakeofsimplicitywewillrestrictourselvestothecaseofT0beingthecharacteristicfunctionofaninterval,T0(x)≡χ[−L,L](x),(1.4)andstudyhowlong-timebehaviorofTdependsonL.Themethodsinthispaperallowonetotreatsomeotherincreasing1-parameterfamiliesofinitialconditions,too.Thus,wewillstudythecompetitionofreactionanddiffusion.Theformerhelpsincreasingthetemperature,whereasthelatter(togetherwiththecompactnessofthesupportoftheinitialdatum)worksto-wardstheextinctionoftheflame.Thisquestionwasoriginallyad-dressedfortyyearsagobyKanel’[9]whoconsideredthecaseofignitionSHARPTRANSITIONBETWEENEXTINCTIONANDPROPAGATION3non-linearityandprovedthatiftheinitialdatumislargeenough,thenreactionwins,whereasifitissmallthendiffusionmanagestoquenchtheflame.Moreprecisely,whenTsolves(1.2)/(1.4)andfisofignitiontype,Kanel’provedthattherearetwolengthscalesL0,L1suchthatT(t,x)→0ast→∞uniformlyinx∈RifLL0,T(t,x)→1ast→∞uniformlyoncompactsifLL1.ThishasbeenextendedtothecaseofbistablefbyAronson-Weinberger[1].Bothresultsactuallyholdwhen(1.4)isreplacedbyT0(x)≡αχ[−L,L](x).(1.5)foranyαθ0,withL0andL1dependingonα(intheignitioncasethisfollowsfrom[9],inthebistablecaseitwasprovedbyFifeandMcLeod[6]).Anaturalquestionarises:doesL0equalL1?IfthisistrueandifonecoulddeterminethebehaviorofTast→∞whenL=L0,thenonewouldbeabletoprovidethecomplete“phaseportrait”forthePDE(1.2)withrespecttoa1-parameterfamilyofinitialconditions.Sincetheseearlyworks,particularlyintherecentyears,severalau-thorshavestudiedquenchingfor(1.1).Theaboveresultshavebeenextendedtothecasewhen(1.1)includesanadvectiontermu·∇T,withuashearorperiodicflow(see[13,15]),evenforcertaincombustionnon-linearities[18].Quenchingoflargeinitialdatabylargeamplitudeshearandcellularflowshasbeenstudiedin[4,5,10,18].However,thequestionwhetherL0=L1remainedopeneveninthesimplestcaseof(1.2).Thefollowingtworesultsprovidetheanswer,includingthetreatmentofthecriticalcaseL=L0.Thefirstofthemholdsforignitionandcombustionnon-linearities.Theorem1.Letθ0∈[0,1)andf:[0,1]→RbeLipschitzwithf(θ)=0whenθ∈[