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Title:MaximalRegularityandAsymptotiBehaviorofSolutionsfortheCahn-HilliardEquationwithDynamiBoundaryConditionsProposedrunninghead:Cahn-HilliardequationAuthors:JanPr ussDepartmentofMathematisandComputerSieneMartin-Luther-University60120HalleGermanypruessmathematik.uni-halle.deReinhardRakeDepartmentofMathematisandStatistisUniversityofKonstanz78457KonstanzGermanyreinhard.rakeuni-konstanz.deSongmuZhengInstituteofMathematisFudanUniversity200433ShanghaiP.R.Chinaszhengfudan.a.nMailingaddress:Prof.Dr.ReinhardRakeDepartmentofMathematisandStatistisUniversityofKonstanz78457KonstanzGermanyVersionofMay27,2003MaximalRegularityandAsymptotiBehaviorofSolutionsfortheCahn-HilliardEquationwithDynamiBoundaryConditionsJanPr ussReinhardRakeSongmuZhengDepartmentofMathematisDepartmentofMathematisandComputerSieneandStatistisMartin-Luther-UniversityUniversityofKonstanz60120Halle78457KonstanzGermanyGermanypruessmathematik.uni-halle.dereinhard.rakeuni-konstanz.deInstituteofMathematisFudanUniversityShanghai200433P.R.Chinaszhengfudan.a.nAbstratThispaperdealswiththeCahn-Hilliardequationt= ; = +3;(t;x)2J ;subjettotheboundaryonditions1 st= s jj gs+h; =0;andtheinitialondition(0;x)=0(x)whereJ=(0;1),and Rnisaboundeddomainwithsmoothboundary =G,n 3,and s; s;gs0;hareonstants.ThisproblemhasalreadybeenonsideredinthereentpaperofRakeandZheng[13℄whereglobalexisteneanduniquenesswereobtained.Inthispaperwe rstobtaintheresultsonmaximalLp-regularityofsolutionandprovethatthesolutionde nesaC0-semigroupinertainSobolevspaes.Wethenstudytheasymptotibehaviorofthesolutionsofthisproblemandprovetheexisteneofglobalattrators.11IntrodutionLet Rnbeaboundeddomainwithboundary = oflassC4,andonsiderthefollowingboundaryvalueproblemfortheCahn-Hilliardequation.t= ; = +3;t0;x2 ;(1.1) =0;t0;x2 (1.2)t s jju+ +gs=h;t0;x2 (1.3)=0t=0;x2 :(1.4)Here (x)denotestheouternormalof atx2 , jjmeanstheLaplae-Beltramioperatoron ,and s; s;gs0;hareonstants.Thisproblemarisesfromthestudyofspinodaldeompositionofbinarymixturesthatappears,forexamples,inoolingproessesofalloys,glassesorpolymermixtures(seeCahnandHilliard[2℄,Novik-CohenandSegal[10℄,Kenzleretal.[8℄,andthereferenesitedtherein.)Boundaryondition(1.3)isusuallyalledthedynamiboundaryonditionsineitalsoinvolvesthetimederivativeofdependentfuntion.Itisderivedwhenthee etiveinterationbetweenthewall(i.e.,theboundary )andtwomixtureomponentsareshort-ranged(seeKenzleretal.[8℄).ThisproblemwasreentlystudiedbyRakeandZheng[13℄andtheglobalexisteneanduniquenessofsolutionwasprovedthere.Furthermore,itwaspointedoutthatfort0thesolutionisC1.However,itwasnotlearwhetherthesolutionde nesaC0-semigroupintheSobolevspaeVintroduedinthatpaper.Inthepresentpaperwefurtherinvestigatethisproblem.Morepreisely,weprovemaximalLp-regularityofsolutionwhihimpliesthatthesolutionde nesaC0-semigroupinertainSobolevspaes.Furthermore,weprovetheexisteneofaglobalattratorforthisproblem.Thispaperisorganizedasfollows.InSetion2,we rststudyalinearproblemassoiatedwithouroriginalproblem(1.1){(1.4),andweestablishmaximalLp-regularityresults.Thenweonsidertheorrespondingnonlinearproblem(1.1){(1.4),andproveinSetions3and4loalwell-posednessaswellasglobalwell-posednessinadi erentphasemanifoldMthanV,whihturnsoutthatthesolutionde nesaC0-semigroupinM.Inthe nalsetionweprovetheexisteneofaglobalattratorinM2(forp=2)aswellasinV.2TheLinearProblemInthissetionwestudythefollowinglinearizedversionof(1.1).tv+ 2v=f;t0;x2 ; v=g;t0;x2 (2.1)1 stv s jjv+ v+gsv=h;t0;x2 v=v0t=0;x2 :Herethefuntionsf,g,haswellastheinitialvaluev0aregiven; s; 0,andgs 0aregivenonstants( =1intheoriginalsystem).2LetJ=[0;T℄and1p1.Wearelookingforsolutionsinthelassv2H1p(J;Lp( ))\Lp(J;H4p( ));whihisthenaturallassfor(2.1)intheLp-setting.Thenbywell-knowntraetheorems(f.[9℄,[1℄,[4℄)thedataf,g,v0neessarilysatisfyf2Lp(J );g2W1=4 1=4pp(J;Lp( ))\Lp(J;W1 1=pp( ));v02W4 4=pp( ):Asusual,hereandinthesequelWspdenotethefrationalSobolevspaes.Furthermore,thetraesofvand von satisfyvj 2W1 1=4pp(J;Lp( ))\Lp(J;W4 1=pp( ));and vj 2W3=4 1=4pp(J;Lp( ))\Lp(J;W3 1=pp( )):Thisleavessomehoieforthesettingofthedynamiboundaryondition.Thepossibilityoflowestorderisthehoieh2Lp(J;W2 1=pp( )).LookingatthedynamialboundaryonditionasaheatequationonJ ,thiswillresultinvj 2H1p(J;W2 1=pp( ))\Lp(J;W4 1=pp( )):Thisregularityimpliesthatthetraeofvj att=0neessarilysatis esv0j 2W4 3=pp( ).Theotherextremepossibilityonsistsintakingtheregularityofthenormalderivativeofvasthebasiregularity,i.e.wemayonsiderthelassh2W3=4 1=4pp(J;Lp( ))\Lp(J;W3 1=pp( )):Thisleadstovj 2W7=4 1=4pp(J;Lp( ))\H1p(J;W3 1=pp( ))\Lp(J;W5 1=pp( )):Thenneessarilyv0j 2W5 3=pp( )andtheompatibilityonditions v0j =gjt=0;forp5;and s jjv0j v0j gsv0j +hjt=02W3 5=pp( );forp5=3;musthold.Moregenerally,anyhoieofthespaeforhofthetypeh2Wsp(J;Lp( ))\Lp(J;Wrp( ));0 s 3=4 1=4p;2 1=p r 3 1=p;willwork.Observethatforsuhr;stheinequality2s risvalid.Theorrespondingtraespaeforv0j nowbeomesWr+2 2=pp( ),andinases1=palsothetimederivativetvj hastraeatt=0,whihbelongstoWr(1 1=sp)p( ).Hereisthemainresultonmaxim