On long-time dynamics for competition-diffusion sy

arXiv:0707.1771v1[math.AP]12Jul2007Onlong-timedynamicsforcompetition-diffusionsystemswithinhomogeneousDirichletboundaryconditionsE.C.M.Crooks,E.N.DancerandD.HilhorstAbstractWeconsideratwo-componentcompetition-diffusionsystemwithequaldiffusioncoefficientsandinhomogeneousDirichletboundaryconditions.Whentheinterspecificcompetitionparametertendstoinfinity,thesystemsolutionconvergestothatofafree-boundaryproblem.Ifallstationarysolutionsofthislimitproblemarenon-degenerateandifacertainlinearcombinationoftheboundarydatadoesnotidenticallyvanish,thenforsufficientlylargeinterspecificcompetition,allnon-negativesolutionsofthecompetition-diffusionsystemconvergetostationarystatesastimetendstoinfinity.SuchdynamicsaremuchsimplerthanthosefoundforthecorrespondingsystemwitheitherhomogeneousNeumannorhomogeneousDirichletboundaryconditions.Keywordsandphrases:Competition-diffusionsystem,boundary-valueproblem,singu-larlimit,long-timebehaviour,spatialsegregation.AMSsubjectclassification:35K50,35B40,35K57,92D25.1IntroductionInthispaper,weshowthat,undercertainconditions,thecompetition-diffusionsystemut=Δu+f(u)−kuvinΩ,vt=Δv+g(v)−αkuvinΩ,(1)withinhomogeneousDirichletboundaryconditionsu=m1≥0on∂Ω,v=m2≥0on∂Ω,(2)hassimplelong-timedynamicsforlargepositivevaluesofthecompetitionparameterk.HereΩ⊂RNissmoothandbounded,fandgarepositiveon(0,1)andnegativeelsewhere,andα0.Suchreaction-diffusionsystemsarewell-knowninthemodellingofcompetitionbe-tweentwospeciesofpopulationdensitiesu(x,t)andv(x,t),andwerefertotheintroduction1of[6]forabriefreview.Thesemodelscanbeusedtostudythedynamicsofthespatialsegregationbetweenthecompetingspecies.Theparameterskandαmaybethoughtofasrepresentingtheinterspecificcompetitionrateandthecompetitiveadvantageofvoverurespectively.Zeroflux(thatis,zeroNeumann)arethemostcommonlyimposedboundaryconditions.Butwhenthetwospecieshavequitedifferentpreferencesforenvironmentalcon-ditions,thencompetitionoccursmainlyinaregionΩwheretheirhabitatsoverlapandthisgivesrisetoboundaryconditions(2)on∂Ω[22].Moreprecisely,weprovethatifαm1−m2isnotidenticallyzeroon∂Ωandallstationarysolutionsofthelimitproblem−Δw=αf(α−1w+)−g(−w−)=:h(w)inΩ,w=αm1−m2on∂Ω,(3)arenon-degenerate(seeDefinition4.1),thenforksufficientlylarge,allnon-negativesolutionsof(1)approachstationarystatesast→∞.Tworemarksonourhypothesesandresultsshouldbemadeattheoutset.First,providedwesupposethatαm1−m2isnotidenticallyzeroon∂Ω,oursystem(1)withinhomogeneousDirichletboundaryconditionshasmuchsimplerdynamicsthanthecorrespondingsystemwithzeroDirichletorNeumannboundaryconditions.In[14],itisobservedthatsuchsystemsmayhavesolutionsthataresmall(O(1/k))foralltime.Toensurethatforlargekthesesolutionsconvergetoastationarysolutionofthek−dependentsystemast→∞,itisnecessarytoimposeaconditionoftherebeingno“circuits”ofpositiveheteroclinicorbitsofanassociatedlimitsystem(see[14,AssumptionC3]and[9])plusaconditiononalinearlimitproblem([14,AssumptionC1]).Nosuchadditionalassumptionsareneededheretoshowsimpledynamics.CompareTheorem4.4with[14,Thm5].Second,theconditionthatallsolutionsofthestationarylimitproblem(3)arenon-degeneratedoesnotalwaysholdforourboundaryconditions-noteveninonespacedimen-sion.ThiscontrastswiththecaseofzeroNeumannorzeroDirichletboundaryconditions,inwhichnon-degeneracydoesholdinonespace-dimension-seearemarkin[14,p472].ButsomegenericityresultscanbeshownforourinhomogeneousDirichletcase,andwediscussthese,togetherwiththepossiblefailureofnon-degeneracyinonedimension,inSection6.Ourmethodsowemuchto[14],whichtreats(1)withzeroNeumannboundaryconditions.Theideaisfirsttouseablow-upmethodtoshowthatforeachδ0,oneofuor/andvmustbesmallateach(x,t)∈Ω×[δ,∞)forsufficientlylargek(Section2).Thisresultsinthelinearcombinationw=αu−vsatisfyingthescalarequationwt=Δw+h(w)+Ok(1),inΩ,(4)w=αm1−m2on∂Ω,wherekOk(1)kL2(Ω)→0ask→∞uniformlyint∈[δ,∞).NotethatherewecanonlyestimatetheL2-normofOk(1),ratherthantheL∞-norm,asin[14],ifthegivenboundarydatam1,m2isnotassumedtobesegregatedon∂Ω.ButthisL2-estimateissufficienttostudythelong-timebehaviourof(1).TheLyapunovfunctionfor(4)withOk(1)=0can2thenbeused(Section3)toshowthatwmustlieclosetosolutionsof(3)fork,tlarge,undertheconditionthatsolutionsof(3)areisolatedinL2(Ω).Section4thenshowsthatifthesestationarysolutionsareinfactallnon-degenerate,thensolutionsof(1)mustapproachstationarystatesof(1)ast→∞.Notethatthenon-degeneracyrequiredinSection4doesimplytheisolatednessusedinSection3,eventhoughthefunctionhin(3)beingonlylocallyLipschitzatitszerosetmeansthattheinversefunctiontheoremcannotbeapplieddirectlytotheoperatorw7→Δw+h(w)(see,forexample,remark(ii)attheendofSection6).Thatthereisa(locally)uniquestationarysolutionof(1)closeto(α−1w+,−w−)forwanon-degeneratesolutionof(3)isshowninSection5usingindex-theoryargumentssimilartothosein[11,10].Ourinhomogeneousboundaryvaluesherenecessitatecarefulmodificationofvariousargumentsin[14,11,10],particularlytheblow-upargumentinSection2,andalsointheboundsandindexargumentsusedtoprovelocaluniquenessinSection5.Section6isdevotedtonon-degeneracyofstationarysolutionsof(3),asmentionedabove.Weuseanapproachfrom[24,8]tos