local solution

applied mathematics presentation and need the explanation and answer to help me learn.

can you prove 3.2 existence and uniqueness not every thing just prove the Local solution
from equation 34 to 38..see the idea just need to prove local solution..also see the note
Requirements:
J.Math.Anal.Appl.412(2014)776Ð791ContentslistsavailableatScienceDirectJournalofMathematicalAnalysisandApplicationswww.elsevier.com/locate/jmaaOnthedynamicsofradiallysymmetricgranulomasAvnerFriedmena,Chiu-YenKaob,RachelLeandera,∗aMathematicalBiosciencesInstitute,TheOhioStateUniversity,Columbus,OH43210,USAbDepartmentofMathematicalSciences,ClaremontMcKennaCollege,Claremont,CA91711,USAarticleinfoabstractArticlehistory:Received24July2013Availableonline12November2013SubmittedbyJ.ShiKeywords:GranulomaSemi-linearparabolicsystemInitialboundaryvalueproblemFreeboundaryvalueproblemAgranulomaisacollectionofmacrophagesthatcontainsbacteriaorotherforeignsubstancesthatthebodyÕsimmuneresponseisunabletoeliminate.Inthispaperwepresentasimplemathematicalmodelofradiallysymmetricgranulomadynamics.Themodelconsistsofacoupledsystemoftwosemi-linearparabolicequationsforthemacrophagedensity,andthebacterialdensity.Theboundaryofthegranulomaisfree.Thissimpleframeworkmakesitpossibletoconductamathematicalanalysisofthesystemdynamics.Inparticular,weshowthatthemodelsystemhasauniquesolution,andthat,dependingonthebiologicalparameters;thebacterialloadeitherdisappearsovertimeorpersists.Weusenumericalmethodstoestablishtheexistenceofstationarysolutionsandexaminehowastationarysolutionchangeswiththereproductiverateofthebacteria.Thesesimulationsshowthatthestructureofthegranulomabreaksdownasthereproductiverateofthebacteriaincreases.2013PublishedbyElsevierInc.1.IntroductionAgranulomaisacollectionofmacrophagesthatcontainsbacteriaorotherforeignsubstances.Granulomasoccurinawidevarietyofdiseasesincluding,forexample,rheumatoidarthritis,schistosomiasisandCrohnÕsdisease.Atypicalexampleisthegranulomaoftuberculosiswhichpreventsresidualbacteriafromre-infectingthebody.Inordertocreateadetailed,disease-speciÞcgranulomamodel,oneneedstoconsider,inadditiontomacrophagesandbacteria,pathogen-speciÞccytokines,theactivationstateofvariousimmunecells,andthedynamicsofbothextracellularandintracellularbacteria.ThiswasdoneinthecaseoftuberculosisbyD.Gammacketal.[1]usingaPDEmodel,byJ.L.Segoria-Juarezetal.[4]usinganagent-basedapproach,andbyS.Marinoetal.[3]usingahybridmulti-compartmentmodel.Inthepresentpaperweintroduceasimplemodelofagenericgranuloma.Themodelexplicitlydescribestheinteractionsbetweenbacteriaandmacrophages.ImplicitinthemodelistheassumptionthatthecytokinesandTcellsarepresentinabundance,i.e.weassumethatallofthemacrophageshavebeenactivatedbyIFN-γsecretedbytheTcells.Similarly,themodeldoesnotconsiderintracellularbacteria,althoughseveraltypesofgranulomas,includingthoseoftuberculosis,arecausedbyintracellularpathogens.WeassumethatthegranulomaoccursinaregionΩ(t)whichvariesintime.InsideΩ(t)themacrophagecelldensity,M,andthebacteriacelldensity,B,satisfyasystemofPDEs.WealsoassumethatthecellulardensityofmacrophageandbacteriaisÞxed,thusourmodeldoesnotaccountfornecroticcellsanddebristhatmaybepresentinseveraltypesofgranulomas.UndertheassumptionthatthecellulardensityisÞxed,thefreeboundaryofΩ(t)moveswithavelocitythatisdeterminedbytheproliferationofthebacteria,theimmigrationofmacrophages,andthedeathofbothcelltypes.*Correspondingauthor.E-mailaddress:rleander@mbi.osu.edu(R.Leander).0022-247X/$Ðseefrontmatter2013PublishedbyElsevierInc.http://dx.doi.org/10.1016/j.jmaa.2013.11.017
A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð791777Theaimofthispaperistoinitiaterigorousmathematicalanalysisofthedynamicsofgranulomasasfreeboundaryproblems.Accordingly,inthepresentpaper,weconsideraverysimplemodelofagenericradiallysymmetricgranuloma,deferringthestudyofmoreinclusivemodelstofuturework.Weprovetheexistenceanduniqueness,andexhibitsteadystatesolutionsnumerically.2.TheoryThevariablexvariesinaboundeddomainΩ(t)inR3withboundaryΓ(t).WeintroducethevariablesM(x,t)andB(x,t)torepresentthedensityofmacrophagesandbacteriarespectively.DuetocellularproliferationanddeaththereisavelocityÞeldv(x,t)whichisassumedtobecommontobothmacrophagesandbacteria.Byconservationofmass,forx∈Ω(t)andt>0wehave∂M∂t−M+∇á(Mv)=−µ1MB−αM,(1)∂B∂t−(1+δ)B+∇á(Bv)=−µ2MB+λB,(2)whereµ1istherateatwhichmacrophagesarekilledbybacteria,µ2istherateatwhichbacteriaarekilledbymacrophages,λisthebacterialgrowthrate,andαistherateatwhichmacrophagesundergoapoptosis.Intracellularbacteriadonotdisperseontheirownbutaredispersedthroughthedispersalofthecellsthatcontainthem,whileextracellularbacteria,beingsmallerthanmacrophages,havealargerdiffusioncoefficientthanmacrophages.Hence,weconsiderthecasewhereδ0;ourresultscanbeextended,withminorchanges,tothecasewhereδ<0.Inaddition,weassumethatthecellsareevenlydistributedinΩ(t)sothat,afternormalization,M+B=1forx∈Ω(t),t>0.(3)AddingEqs.(1)and(2)andusing(3),wederivethefollowingequationforv:∇áv=−δM+λ−(λ+µ+α)M+µM2,(4)whereµ=µ1+µ2.Inaddition,replacingBwith1−Min(1)yieldsthefollowingequationforM:∂M∂t−M+∇á(Mv)=−µ1M(1−M)−αM.(5)Inthispaperweconsideronlythecaseofradiallysymmetricgranulomas.Inthiscasevisdeterminedby(4)togetherwithv(0)=0.Inthenon-radiallysymmetriccaseonewouldneedtoimposeaconstitutiveconditiononthetissuewherethegranulomadevelops.SuchaconditioncouldbetheporousmediumassumptioncharacterizedbyDarcyÕsLaw:v=∇p,wherepistheinternalpressure,andpsatisÞesanappropriateboundaryconditionontheboundaryΓ(t).Thismoregeneralgranulomamodelcouldbeconsideredinfuturework.ItiseasilyseenthatifMsatisÞes(5)withvdeÞnedby(4),thenthepair(B,M)satisÞesthesystem(1)Ð(2).Inthesequelweshallprimarilyusetheversion(4)Ð(5)ofthesystem(1)Ð(3).Weimposetheboundaryconditions∂M∂ν=β(1−M)onΓ(t),(6)vΓ(t)=váνonΓ(t),(7)whereνistheoutwardnormaldirection,vΓ(t)isthevelocityofthefreeboundary,Γ(t),inthedirectionν,andβ>0.Finally,weprescribeinitialconditions:Ω(t)|t=0=Ω0,M(x,0)=M0,0M01.(8)Notethat(6)implies(by(3))that∂B∂ν+βB=0onΓ(t).(9)Inaddition,(8)impliesthat0B(x,0)1.(10)Bythemaximumprinciplefor(1)and(2)wethenhavethatM(x,t)0andB(x,t)0.Thusby,(3),thesolutionof(4)Ð(5)satisÞes0M(x,t)1.(11)
778A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð7912.1.TheradiallysymmetriccaseWerewritethesystemintheradiallysymmetriccase.Usingthenotationr=|x|,v=v(r,t)xr,M=M(r,t),Ω(t)=r0,R(t)R0eCT,−c0eCTdRdtCR0eCTfor00.ThenwsatisÞes:∂w∂t−(1+δM)∂2w∂r2+2r∂w∂r−δN∂w∂r+v∂w∂r+(1+δM)2r2w+(f+γ)w=2δre−γtw2,and,bythemaximumprinciple,wcannottakeanegativeminimuminthedomain{00.Applyingthisinequalitytou=öM−M0andto∂u∂r,weget|öM|Cα,α2+|öMr|Cα,α22|M0|C2+α+c0TγLforsomepositiveconstantsc0andγ.HenceifL2=2|M0|C2+α+1andTissufficientlysmallthenöM=S(M)∈YT.SimilarlywecanshowthatSisacontraction.TheproofisobtainedbyestimatingthedifferenceöM1−öM2fromEq.(34)forM1andM2,usingtheestimates:v(M1)−v(M2)Cα,α2c1M1−M2,c(M1)−c(M2)Cα,α2c2M1−M2.SinceSisacontractiononYTforTsufficientlysmall,givenR(T)∈XTthereexistsauniquesolutionM(r,t)of(12)Ð(15).WeproceedtosolveforöR:döRdt=vMR(t),t,t,öR(0)=R0anddeÞneamappingVbyVR=öR.WeclaimthatVisacontractiononXTandthushasauniqueÞxedpoint.ItwillbeconvenienttoconsidertheequivalentÞxedboundarysystem(23)Ð(27).GivenR1andR2letM1,v1andM2,v2solvethecorrespondingsystem(23)Ð(26).ToestimateM1−M2wemoveallthetermswithR1−R2,úR1−úR2totheright-handsideandapplytheSchauderestimatestoget|M1−M2|C2+α,1+α2C|úR1−úR2|Cα2.Byinterpolation,asbefore,|M1,r−M2,r|Cα,α2c1Tγ|M1−M2|C2+α,1+α2c0CTγ|úR1−úR2|Cα2.(37)Lett10,theestimate,|úR|Cα2tL0(39)foranysolutionwhichexistsfor0tTwithR(T)>0,whereL0=L0(T)isaboundedfunctionofT.ByLemma1ifδ=0thenMrisboundedbyaconstantøLandthensoisv.HencewecanusetheW2,pestimatetoconcludethatM,∂M∂t,∂M∂ρ,∂2M∂ρ2areboundedinLpuniformlyintbyanotherconstantøL.Itfollowsthat∂M∂ρCα,α2øL.(40)Proceedingsimilarlyto(38)wededucethatúR(t2)−úR(t1)c|t2−t1|α2.Insummary:Theorem2.Undertheassumption(33)thereexistsauniqueglobalsolution(i.e.for0t0,weareunabletoruleoutthepossibilitythatR(t)remainsuniformlyboundedbyapositiveconstantast→T∞,whileMr(rj,tj)→∞forsequences00,and∂w∂r+βw=βw>0onr=R(t).Hence,bythecomparisonprincipleforparabolicequationsB(r,t)0(asinTheorem3)andissufficientlysmallthenfrom(17)and(14)wegetúR=vR(t),t−α3+O()R(t).HenceúR<0.Furthermorein(32)f=σ+O(),whereσ=α3−γ,canbepositiveornegative.If∂M0∂rηwhere(1+δ)2r2+σ−2δrη>0for0µ2andβ=0thenR(T)0r2B(r,T)drR00r2B(r,0)dr+T0R(t)0r2(λ−µ2)B(r,t)drdt.(47)for0Tµ2,Theorem4showsthatthebacterialloaddoesnotdecreaseandthegranulomagrows.Theseresultssuggestthatforβ=δ=0andµ2<λ<µ2+αthereshouldbestationarygranulomaswheremacrophagesandbacteriacoexist.InthenextsectionweexhibitstationarysolutionsnumericallyandalsodiscusshowgranulomassatisfyingtheconditionsofTheorems3and4evolveovertime.Theorem5.If∂M0∂r0,λ>µ2,δ=0,R(0)>3βλ−µ2,(51)andR00r2(λ+µ+α)B(r,0)dr>R303(µ+α)(52)thenR(T)0r2B(r,T)drR00r2B(r,0)dr+T0R(T)0r2µ2B2(r,t)drdt(53)andR(T)>3βλ−µ2,for0T0(55)impliesR(t)0r2B(r,t)dr>R00r2B(r,0)dr+T0R(t)0r2µ2B2(r,t)drdt.(56)Henceitsufficestoshowthat(55)holdsforallt>0.SinceR(0)>3βλ−µ2,(57)if(55)doesnotholdforallt0suchthat(55)holdsforallt3βλ−µ2fort∈[0,T∗].Sinceδ=0,R(t)=vR(t),t1R2(t)R(t)0r2(λ+µ+α)B(r,t)−(µ+α)dr,(59)andbyassumption(52),R(0)>0.Hence,thereexistsan>0sothat
786A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð791R(t)>R0>3βλ−µ2(60)fort∈[0,).WeclaimthatR(t)>R0foralltR0,(61)whichisacontradiction.Hence,R(t)>R0>3βλ−µ2fort∈[0,T∗).ItfollowsthatT∗0(λ−µ2)3RT∗−βR2T∗BRT∗,T∗dt>0,acontradictionto(58).Wehavethuscompletedtheproofof(55)and(56)and,atthesametime,establishedtheestimateR(t)>3βλ−µ2.✷4.1.NumericalsimulationsInonedimension,themovingboundaryproblemcanbesolvedbymappingthemovingdomain[0,R(t)]intotheÞxeddomain[0,1]byρ=rR(t);see(23)and(24).∂M∂t+v−ρúR(t)R(t)−2ρR(t)2∂M∂ρ−1R2∂2M∂ρ2=−2vρRM−1R∂v∂ρM+E(M),∂∂ρρ2v=−δR∂∂ρρ2∂M∂ρ+ρ2RF(M),(62)whereE(M)=−µ1M(1−M)−αMandF(M)=λ−(λ+µ+α)M+µM2.Letusdenotethenumericalsolutionatthen-thtimestepbyMnj,Vnj,Rnjatxj=(j−1)h,1jJwith(J−1)h=1.WeÞrstcomputeVn+1bythetrapezoidalrule:ρ2j+1Vn+1j+1−ρ2jVn+1j=Mc+Rn2ρρ2j+1Fnj+1+ρ2jFnj,whereMc=−δRρ2j+1Mnj+2−Mnj2ρ−ρ2jMnj+1−Mnj−12ρforjJandMc=−δRρ2j+1Rnβ1−Mnj−ρ2jMnj+1−Mnj−12ρforj=JisthecentralschemeapproximationfortheÞrsttermontheright-handside.TheEulermethodisusedtoupdatetheradius:Rn+1=Rn+tVn+1(1,t).
A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð791787WewritetheadvectionÐdiffusionÐreactionequationinthefollowingform:∂M∂t+A1∂M∂ρ+A2∂2M∂ρ2=r(M,v,R)M,whereA1=v−ρúR(t)R(t)−2ρR(t)2,A2=−1R2,andr(M,v,R)=−2vρR+1R∂v∂ρ+µ1(1−M)+α.WeusetheschemeMn+1j−Mnjt+(A1)n+1jMn+1j+1−Mn+1j−12ρ+(A2)n+1jMn+1j+1−2Mn+1j+Mn+1j−1(ρ)2=rMnj,Vn+1j,Rn+1Mn+1j,(63)wherethederivativeterm,∂v∂ρ,inr(M,v,R)isapproximatedbytheforwarddifference,i.e.∂v∂ρ≈Vn+1j+1−Vn+1jρ.Thisdiscretizedequation(63)canbewrittenintheformbjMn+1j−1+djMn+1j+ajMn+1j+1=Sj,j=2,…,J−1,whereaj=(A1)n+1jρ+2(A2)n+1j(ρ)2t,bj=−(A1)n+1jρ+2(A2)n+1j(ρ)2t,dj=2−22(A2)n+1j(ρ)2+rMnj,Vn+1j,Rn+1t,Sj=2Mnj.Forj=1,wehavetheboundaryconditionMn+10=Mn+12,whichimpliesthata1=4(A2)n+11(ρ)2t,d1=2−22(A2)n+11(ρ)2+rMnt,S1=2Mn1.Forj=J,wehavetheboundarycondition1Rn+1Mn+1J+1−Mn+1J−12ρ=β1−Mn+1j.Thus,bJ=4(A2)n+1j(ρ)2t,dJ=2−221+ρβRn+1(A2)n+1j(ρ)2+rMn+(A1)n+1JβRn+1t,SJ=2Mnj−2t(A2)n+1J2ρβRn+1+(A1)n+1JβRn+1.
788A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð791Fig.1.TheintersectionpointsforΓ1andΓ2forλ=0.375:0.125:1,µ1=0.4,µ2=0.6,δ=0.05,α=0.5,β=0.8.Asλincreases,theradius,R,decreasesandtheconcentrationofmacrophagesatthecoreofthegranuloma,M0,increases.ToÞndthestationarystatesolution,i.e.Mandvwhichsatisfy∂2M∂r2−M∂v∂r=v−2r∂M∂r−E(M)+2rvM,∂v∂r=11+δMδE(M)+F(M)−δv∂M∂r−2(1+δM)vr,weÞrstwritetheproblemasasystemofÞrst-orderequationsu1u2−u1u3u3=u2(u3−2r)u2+2ru3u1−E(u1)11+δu1[δE(u1)+F(u1)−δu3u2−2(1+δu1)u3r]whereu1=M,u2=∂M∂r,u3=v,andtheinitialconditionsareu1(0)u2(0)u3(0)ρ=0=M000.Foragivensetoftheparametersδ,λ,µ,α,andβ,wesolvethesystemofequationsintheintervalr∈[0,R]withvariousM0andRandthenÞndtheintersectionpointsofthecurvesΓ1andΓ2,whichsatisfyΓ1:u2(1)−β1−u1(1)=0(boundaryconditionsforM),Γ2:u3(1)=0(stationaryfreeboundary).Intersectionpointsofthesetwocurvesrepresentstationarysolutions.AfterÞndinganintersectionpoint,(÷M0,÷R),wecanthenÞndthecorrespondingsolutionM=u1andv=u3.SomestationarypointsandthecorrespondingsolutionsareshowninFigs.1and2.Numericalsimulations(notshownhere)indicatethatthestationarysolutionsshowninFigs.1and2arenotstable.FortheparametervaluessatisfyingtheconditionsofTheorem4thereisnostationarysolutionsincelimsupt→∞R(t)=∞.InthesimulationsofFigs.1and2theparameterβisnonzero,sothattheconditionsofTheorem4arenotsatisÞed.
A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð791789Fig.2.Thecorrespondingstationarysolutionsforλ=0.375:0.125:1,µ1=0.4,µ2=0.6,δ=0.05,α=0.5,β=0.8.AsλincreasesthestructureofthegranulomabreaksdowninthatmacrophagesinÞltrateitscore.Itisinterestingtoconsiderhowstationarysolutionschangeasafunctionofthebiologicalparameters.Figs.1and2illustratehowthelocationofastationarypoint(in(R,M)space)andthecorrespondingstationarysolutionchangeasafunctionofλ(thereproductiverateofthebacteria).Whenλissmall,bacteriaareconcentratedatthegranulomaÕscorewhilemacrophagesareprimarilyfoundatitsboundary.Asλincreasesthegranulomabecomessmallerandlessstructuredwithmacrophagesdistributedthroughout.Weinterpretthistomeanthatthefastergrowingbacteriarequireagreaterconcentrationofactivatedmacrophagesforcontainment.Highermacrophageconcentrations,inturn,resultinasmallergranulomawithalowerbacterialload.NextwecomparetimedependentsolutionsthatsatisfytheconditionsofTheorem3tothosethatsatisfytheconditionsofTheorem4.InFig.3,weshowtheevolutionofthetimedependentsolutionofthesystem(62)withparametersthatsatisfytheconditionsofTheorem3for=0.25:λ=0.5,µ1=0.4,µ2=1.5,δ=0.05,α=0.5andβ=0.8.TheinitialradiusischosenasR(0)=4.5374andtheinitialmacrophageconcentrationischosenasM=0.75+0.25ρ2.Astimegoeson,themacrophageconcentrationbecomesoneeverywhere,andtheradiusofthegranulomashrinks.InFig.4,weshowtheevolutionofthetimedependentsolutionofthesystem(62)withparametersthatsatisfytheconditionsofTheorem4(λ=2.7,µ1=0.4,µ2=1,δ=0.05,α=0.5andβ=0.8).TheinitialradiusischosenasR(0)=4.5374andtheinitialmacrophageconcentrationischosenasM=0.75+0.25ρ2whichisthesameastheexampleinFig.3.Astimegoeson,themacrophageconcentrationdecreasestozeroeverywhereexceptforasmallregionnearthegranulomaÕsboundary,and,incontrasttotheexampleshowninFig.3,theradiusofthegranulomaincreases.5.ConclusionsInthispaperweinitiatedastudyofasimplemathematicalmodelofagenericgranuloma.Themodelconsistsofacoupledsystemoftwosemi-linearparabolicequationsforthemacrophagedensity(M),andthebacterialdensity(B).Theboundaryofthegranulomaisafreeboundary.Weprovedtheexistenceanduniquenessofasolutionandproceededtoex-plorehowgranulomasevolve.Dependingonthebiologicalparameters,weshowedthatthebacterialloadeitherdisappearsovertime(Theorem3andRemark1)orpersists(Theorems4and5).Wehavealsoshownnumericallythatthereexistunstablestationarysolutions.Fivebiologicalparametersdetermineifthebacterialloadandgranulomawillgroworshrink:Thenaturaldeathrateofthemacrophages(α),thegrowthrateofthebacteria(λ),theßuxofthemacrophagesfromthehealthytissueintothegran-uloma(β),therateatwhichbacteriakillmacrophages(µ1),andtherateatwhichmacrophageskillbacteria(µ2).Although
790A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð791Fig.3.Thetimedependentsolutionforλ=0.5,µ1=0.4,µ2=1.5,δ=0.05,α=0.5,β=0.8,andR(0)=4.5374.Fig.4.Thetimedependentsolutionforλ=2.7,µ1=0.4,µ2=1,δ=0.05,α=0.5,β=0.8,andR(0)=4.5374.
A.Friedmenetal./J.Math.Anal.Appl.412(2014)776Ð791791themodelconsideredhereisasimpliÞcation,itisabletocapturecertainfeaturesofrealworldgranulomas.Indeedmath-ematicalanalysisandnumericalsimulationsofthemodelindicatethat,asinthegranulomasoftuberculosis,macrophagesaremoreprevalentatthegranulomaÕsedge.Thestructureofstationarygranulomasappearstodeteriorate,however,asthebacterialgrowthrateincreases.Thisobservationisofspecialinterestsinceunstructuredgranulomasareahallmarkofactivetuberculosisinfections.Futureworkshould(i)betterdetermineparameterregimesinwhichmacrophagesandbac-teriacoexist;(ii)rigorouslyestablishtheexistenceofstationarysolutionsaswellasanalyzetheirasymptoticstability;and(iii)includemorerealisticmodelsofgranulomas.AcknowledgmentsChiu-YenKaoispartiallysupportedbyNSFDMS-1318364.TheworkofRachelLeanderwaspartiallysupportedbytheNationalScienceFoundationunderAgreementNo.0931642.References[1]D.Gammock,C.Doering,D.Kirschner,MacrophageresponsetoMycobacteriumtuberculosisinfection,J.Math.Biol.48(2004)218Ð242.[2]G.M.Lieberman,SecondOrderParabolicDifferentialEquations,WorldScientiÞcPublishingCo.,1996.[3]S.Marino,M.El-Kebir,D.Kirschner,Ahybridmulti-compartmentmodelofgranulomaformationandtcellprimingintuberculosis,J.Theoret.Biol.280(2011)50Ð62.[4]J.L.Segovia-Juarez,S.Ganguli,D.Kirschner,IdentifyingcontrolmechanismsofgranulomaformationduringM.tuberculosisinfectionusinganagent-basedmodel,J.Theoret.Biol.231(2004)357Ð376.