Asymptotic behavior of solutions of an Allen-Cahn

DépartementdeMathématiquesetInformatiqueCNRSURA762________________________________________________UPERIEURESORMALENECOLEAsymptoticBehaviorofSolutionsofanAllen-CahnEquationwithaNon-localTermXinfuCHENDanielleHILHORSTElisabethLOGAKLMENS-95-4AsymptoticBehaviorofSolutionsofanAllen-CahnEquationwithaNon-localTermXinfuCHENDanielleHILHORSTElisabethLOGAKLMENS-95-4March95LaboratoiredeMathematiquesdel’EcoleNormaleSuperieure45rued’Ulm75230PARISCedex05Tel:[phone]Adresseelectronique:[e-mail]s,UniversityofPittsburghLaboratoired’AnalyseNumerique,CNRSUniversiteParis-Sud(B^at425)91405ORSAYCedexAsymptoticbehaviorofsolutionsofanAllen-Cahnequationwithanon{localtermXinfuChenxDanielleHilhorst{ElisabethLogakkAbstractWeconsideranAllen-Cahnequationinvolvingaspatialaverageterm,whicharisesasasingularlimitofareaction-diusionsystemmodellingphasetransi-tion.Weshowthatthesolutionofthisequationconvergestoastepfunctiontakingvalues1onbothsidesofamovinginterface.Thenormalvelocityoftheinterfaceisgivenbythesumofmeancurvatureandofsomenon-localtermdependingonthevolumedelimitedbytheinterface.Weproveexistenceanduniquenessofasmoothsolutionofthisfree-boundaryproblemlocallyintime.KEYWORDS.Allen-Cahnequation-Integro-dierentialequations-Singularlimit-Propagationofinterfaces.1Introduction.Inthispaper,westudythelimitingbehavior,as&0,ofthesolutionuofthefollowingproblem(P)8:ut=u+12f(u;Zu)inIR+;@u@n=0on@IR+;u(x;0)=g(x)x2;(1.1)whereIRN(N2)isasmoothboundeddomainandnistheunitoutwardnormalto@.Weassumethatf(u;v)issmoothandthat~f(u)=f(u;0)istheTheauthorsareverygratefultoProfessorY.Nishiuraformanystimulatingdiscussions.xDepartmentofmathematics,UniversityofPittsburgh,Pittsburgh,PA15260,USA.PartiallysupportedbytheNationalScienceFoundationGrantDMS{9200459andtheAlfredP.SloanResearchFellowship.PartofthisworkwasdoneduringavisittotheUniversityofParis-Sud.{Laboratoired’AnalyseNumerique,CNRSetUniversiteParis-Sud,B^atiment425,91405Orsay,FRANCE.kDepartementdeMathematiquesetd’Informatique,EcoleNormaleSuperieure,45,rued’Ulm75230ParisCedex05,FRANCE.1oppositeofthederivativeofadoubleequalwellpotentialtakingitsglobalminimumvalue0atu=1;moreprecisely,weassumethat~f(u)hasthefollowingproperties:~f(1)=0;~f0(1)0;Z11~f(s)ds=0(1.2)andthereexistsauniquevaluea2(1;+1)suchthat~f(a)=0(1.3)with~f0(a)0:(1.4)Atypicalexampleis~f(u)=uu3andf(u;v)=~f(u)+vh(u).Whenf(u;v)isindependentofv,i.e.,f(u;v)=~f(u),equation(??a)reducestotheAllen{Cahnequation[?]modellinganantiphaseboundarymotionandantiphasedomaincoarsening.ThemainfeatureofthesolutionoftheAllen{Cahnequationisthatthezerolevelsetofthesolutionapproximates(as!0)themotionbymeancurvatureow.FormalderivationofthisconnectionwascarriedoutbyAllen{Cahn[?],Rubinstein,Sternberg,andKeller[?],Fife[?]andmanyothers.Rigorousjusti-cationwassuccessfullycarriedoutrecentlybyBronsardandKohn[?],deMottoniandSchatzman[?][?],Chen[?],Evans,SonerandSouganidis[?],Ilmanen[?]andmanyothers.Problem(P)isobtainedasthelimit,as&0and&0,ofthefollowingsystemofreaction-diusionequations(ut=u+12f(u;jjv)inIR+;vt=1v+u1vinIR+(1.5)withappropriateinitialandboundaryconditions,whereandarexedpositiveconstants.(Seex4forarigorousjustication.)Thesystem(??)isusuallyreferredasaBelousov{Zhabotinskiimodel(orFitzHugh{Nagumomodelwhen=1)whichdescribes,forinstance,wavepropagationinexcitablemedia,patternformationinpopulationgenetics,propagationofsignalsalonganerveaxonorcardiactissue,etc,andhasbeenextensivelystudied;see,forexample,NishiuraandMimura[?],OhtaandKawasaki[?],FifeandTyson[?],Fife[?],andthereferencetherein.Forxedpositiveconstants;;and,thelimitingbehavior,as!0,ofsolutionsoftheparabolicsystem(??)wasrecentlystudiedbyX.Chen[?]andX-Y.Chen[?].Itiswell{knownthatthemotionbymeancurvatureowdoesnothavenon{trivialstationarypatternsthatdonotattach@.Infact,incase=IRN,everyinitiallyboundedhypersurfaceshrinksintopointsinnitetime;see,forexample,EvansandSpruck[?].However,forequation(??a),thereisanon{localtermrepresentedbytheintegralofu,whichprovidesdynamicsfortheformationofnon{trivialstationarypatterns.Hence,thesolutionsofProblem(P)providesricherstructurethanthatoftheAllen{Cahnequation.2Wenowstatethemainresultofthispaper.Theorem1.1.Assumethatf(u;v)2C2([2;2][1;1])andthat~f(u):=f(u;0)satises(??).Let0beagivensmoothclosedhypersurfacewhichseparatesintotwodomains+0and0.Let:=[0tTtftgbeafamilyofsmoothhypersurfacesthatmovesaccordingtothelaw(V=K+c0(j+tjjtj);tjt=0=0(1.6)whereVisthenormalvelocityoft(positiveiftisexpanding),Kisthemeancurvatureoft(positiveiftisconvex),c0iscertainconstantthatdependsonlyonf(cf(??)inx3),tistheregionenclosedbyt,+t=n(t[t),andjAjisthemeasureofthesetA.Assumethattforall0tT.Thenthereexistsafamilyofcontinuousfunctionsfgg01suchthatthesolutionuofproblem(P)withinitialdatagsatiseslim!0u(x;t)=18x2t;t2[0;T]:Actually,astrongerresultthanTheorem1.1willbeproved.Theideaoftheproofistoconstructsub{supersolutionpairsandtouseacomparisonprinciple.Weshallestablishthee