A STUDY OF GLOBAL EXISTENCE TOWARD SOME CHEMOTAXIS MODELS By Minh Le A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics – Doctor of Philosophy 2024 ABSTRACT The focusofthisdissertationisonglobalexistenceofsolutionsofsomechemotaxissystemswith logistic sources,subjecttobothhomogeneousandnonlinearNeumannboundaryconditions.It is knownthatblow-upphenomenacanbepreventedforsomechemotaxismodelswithquadratic logistic sourcesunderhomogeneousNeumannboundaryconditions.However,itisshowninthis research thatquadraticlogisticsourcesarenotoptimalforpreventingblow-upphenomenainsome chemotaxis systems.Indeed,ourfirstresultdemonstratesthatKeller-Segelsystemscanavoid blow-up solutionsundernonlinearNeumannboundaryconditionswithquadraticlogisticsources. Moreover,thesecondresultshowsthatblow-upsolutionscanbepreventedintwospatialdimen- sions withsub-logisticsources.Additionally,thethirdresultshowsthatsub-logisticsourcesare even sufficienttoavoidblow-upsolutionsundernonlinearNeumannboundaryconditionsintwo spacial dimensions.Furthermore,weinvestigatesomenonlinearnonlocalsourcesKellersystems and theeffectofsub-logisticsourcesundernonlinearNeumannboundaryconditionsandtwo- species withtwochemicalsmodelsintwospacialdimensions.Finally,weshowthatthepresence of logisticsourcescanpreventblow-upphenomenoninsuperlinearcrossdiffusionchemotaxisand in superlinearsignalproductionchemotaxismodels.Mathematicaltoolsutilizedintheresearch include variationalmethods,Moser-Alikakositerations,regularitytheoryforellipticandparabolic equations inSobolevspacesandinOrliczspaces,someelementalinequalitiesinSobolevspaces and differentialinequalities. Copyright by MINH LE 2024 ACKNOWLEDGEMENTS I wouldliketoexpressmyheartfeltgratitudetomydearwife,ThaoVyHoangNguyen,forher unwavering loveandsupportthroughoutmyPhDjourney.Withouther,thisachievementwould not havebeenpossible.Ialsoextendmythankstomyfather,mother,andbrotherfortheirsupport and trustinmetocompletethiswork. My sincerestappreciationgoestomyadvisor,Prof.ZhengfangZhou,whoseopen-mindedness and broadvisioninspiredmetopursuethisresearchtopic.Heprovidedmewithinvaluableguid- ance andencouragementthroughouttheprocess,dedicatinghistimetoverifymyworkandengage in manydiscussions. I amalsogratefultoProf.JunKitagawa,whoservedonthecommitteeandwhoselectureon parabolic regularitytheorylaidthegroundworkformyresearch,specificallytheDeGiorgiMoser iteration technique.Furthermore,IextendmythankstoProf.BaishengYan,Prof.WilleWong, and Prof.OlgaTuranovafortheirparticipationinthecommittee. Additionally,IamdeeplygratefultoProf.MichaelWinklerforhisvaluablecomments,sug- gestions, anddiscussionsinChapter2,Chapter3,andChapter5. Lastly,Iwishtoexpressmywarmappreciationtomyanonymousfriendsfortheirhelpful suggestions andcomments. iv TABLEOFCONTENTS CHAPTER 1INTRODUCTION . ............................1 CHAPTER 2CHEMOTAXISWITHLOGISTICSOURCESUNDERNONLINEAR NEUMANN BOUNDARYCONDITION . ................6 CHAPTER 3BLOW-UPPREVENTIONBYSUB-LOGISTICSOURCESIN2D KELLER-SEGEL SYSTEM . .......................32 CHAPTER 4NONLOCAL;TWOSPECIESWITHTWOCHEMICALS;ANDNON- LINEAR BOUNDARYPROBLEMS . ..................48 CHAPTER 5SUPERLINEARCROSS-DIFFUSION;SUPERLINEARSIGNAL PRODUCTION . .............................80 BIBLIOGRAPHY . .......................................112 APPENDIX ANOTATIONS . ..............................118 APPENDIX BINEQUALITIES . ............................121 APPENDIX CREGULARITYTHEORY . .......................127 v CHAPTER 1 INTRODUCTION My inspirationforthisprojectstemsfromvariouspapersandabook[50]. Itisdiscoveredthat cross-diffusionsystems,suchasKeller-Segel,playasignificantroleinpredictingtheformationof aggregations, navigatinganoptimalpathinacomplexnetwork,andeveninphysics,suchasparticle interaction. Manymathematicaltoolsofnonlinearparabolicequationtheorycanbeadaptedand modified totacklechemotaxissystemslikeKeller-Segelanditsvariations.Asaresult,thisthesis aims toinvestigatesolutions,includingglobalexistenceofcertainchemotaxissystemsusingthese techniques. The dissertationisstructuredasfollows:Chapter2examineschemotaxismodelswithlogistic sources undernonlinearNeumannboundaryconditions.InChapter3,weanalyzeachemotaxis model withsub-logisticsourcesintwospatialdimensions.Inchapter4,weinvestigatetheglobal existence issuesofseveralchemotaxissystemsincludingKeller-Segelsysteminthelimitingpa- rameter,twospecieswithtwochemicalssystems,nonlocalsourceschemotaxissystemsinany spacial dimensions n ≥ 3. Furthermore,intwospacialdimensionsthepresenceofsub-logistic sources canpreventblow-upevenforchemotaxissystemundernonlinearNeumannboundarycon- ditions. Chapter5isdevotedtoinvestigatingglobalexistenceissuesintwochemotaxismodels: one featuringsuperlinearcross-diffusionratesandsub-logisticsources,andanotherincorporating superlinear signalproductionwithlogisticsources.Finally,inappendix,weadoptandmodify Moser-AlikakositerationmethodforgeneralclassofchemotaxissystemswithgeneralNeumann boundary condition. Each chapteriswrittenasaself-containedunit,soreaderscanapproachthemindependently and donotnecessarilyneedtoreadtheminorder.Thisstructureallowsreaderstofocusonspecific areas ofinterestandgaininsightsintoparticularaspectsofchemotaxismodelswithouthavingto read theentiredissertation. 1 1.1 Somebackgroundofchemotaxissystems The chemotaxisphenomenon,whichreferstothemovementofcellstowardchemicalsignal, has beenintensivelystudiedsince1880s.Howeveruntil1970s,themathematicalmodellingfor chemotaxis wasfirstintroducedin[26]. Tobemorespecific,thegeneralformofthemodelis described bythefollowingPDEs: 8>>< >>: ut = ∇ · (D(u; v)∇u − S(u; v)u∇v) + f(u; v) vt = dΔv + g(u; v) − h(u; v)v (1.1.1) where u(x; t); v(x; t) are functionscorrespondinglydescribingthecelldensitypopulationand chemical signalconcentrationataposition x in anopensmoothboundeddomain Ω ⊂ Rn and a instantoftime t. Thefunctions D(u; v) and S(u; v) represent diffusivityofthecellsandthe chemotactic sensitivity,respectively.Thefunction f(u; v) describes cellgrowthanddeathwhile the function g(u; v) and h(u; v) are kineticfunctionsthatdescribeproductionanddegradationof chemical signal,respectively.Asteptowardthederivationofthemodelaswellasmanyinterest- ing resultscanbefoundin[20]. Also,manydevelopedtechniquesandresultsoverdecadestodeal with thechemotaxissystemshasbeensummarizedin[4]. Moreover,[18] presentsanextensive survey ofvariouschemotaxismodels,alongwiththeircorrespondingbiologicalbackground.Fur- thermore, [17] offersasummaryofkeytechniquesandrecentfindingsregardingglobalexistence and blow-upsolutions,aswellasadiscussionofrecentnumericalstudiesandpotentialdirections for futureresearch. Since the1970s,themathematicalperspectiveofthisphenomenonhasrapidlyevolved,fueledbyits crucial applicationsandinherentmathematicalelegance.Amongthemanyareasofresearchinthis field, oneofthemostfascinatingaspectsofchemotaxissystemsisthecriticalmassphenomenon. Specifically,when S ≡ D ≡ 1, g(u; v) = u, h ≡ 1,  = 0, d = 1, and Ω = R2, thesimplicity and eleganceofthissystemhavecapturedtheattentionofnumerousresearchers,intriguedbythe critical massphenomenonitexhibits(seee.g.[12, 7, 42, 43, 52, 51, 45]...). Heuristically,the analysis pictureofthiscriticalcanberoughlyunderstoodbydifferentiatingthesecondmomentum of thesolution(see[19]). Specifically,thesecondequationof(1.1.1), thePoissonequation,gives 2 us anexplicitformulafor v in termsof u. v(x; t) = − 1 2 Z R2 ln(|x − y|)u(y) dy By substitutingthisintotheevolutionequationfor u, weobtain: ut = Δu + 1 2 ∇ · (u∇ln |x| ∗ u) (1.1.2) Wecancalculatethedissipationofthesecondmomentumexplicitlybyusingintegrationbyparts d dt Z R2 |x|2u(x; t) dx = Z R2 |x|2 (Δu − ∇· (u∇v)) dx = Z R2 Δ(|x|2)u dx + Z R2 (2x · ∇v)u dx = 4m + 2 Z R2 (x · ∇v)u dx = 4m − 1  Z R2 Z R2 u(x; t)u(y;t) x · (x − y) |x − y|2 dxdy = 4m − 1 2 Z R2 Z R2 u(x; t)u(y;t) (x − y) · (x − y) |x − y|2 dxdy = 4m − m2 2 ; where m = R R2 u(x; t) dx. Iftheinitialsecondmomentumisfinitewehave Z R2 |x|2u(x; t) dx = Z R2 |x|2u(x; 0) dx + 4m(1 − m 8 )t: As aconsequence,wefindthatsolutionsdonotexistgloballywhen m > 8. Indeed,wehavea rich literatureconcerningaboutthiscriticalmass m = 8. Formoredetails,interestedreadersare referred to[19][Introduction]. In additiontothebiologicalchemotaxisphenomena,thelogisticterm f(u) = au−u2 introduced in theevolutionequationfor u plays aroleindescribingthegrowthofthepopulation.Specifically, the term au, with a ∈ R, isthegrowthrateofpopulationandtheterm −u2 models additional overcrowding effects.Inatwo-dimensionalspace,thesystem(KS) possessesauniqueclassicalso- lution whichisnonnegativeandboundedin Ω×(0;∞) ( seee.g.[47, 46]). Inahigher-dimensional space, thesimilarresultscanbefoundin[21] fortheparabolic-ellipticmodel,andin[65] forthe 3 parabolic-parabolic modelwithanadditionallargenessassumptionof . Inadditiontotheglobal classical solutions,theexistenceglobalweaksolutionsresultsforanyarbitrary  > 0 were also obtained in[21] fortheparabolic-ellipticmodelsandin[28] fortheparabolic-parabolicmodelsin a three-dimensionalsystem.Furthermore,apreciseformulaforalowerboundfor  was foundin [69]. 1.2 Problemsandresults The maingoalofthisthesisistoinvestigatetwocrucialinquiriesconcerningtheglobalbound- edness ofsolutionsincertainchemotaxissystems. The firstquestionrevolvesaroundtheglobalexistenceofsolutionsundernonlinearNeumann boundary conditions,suchas ∂u ∂ν = |u|p for p > 1. InChapter2,weestablishtheglobalexistence of solutionswhen p is belowacriticalthreshold.Specifically,Theorem 2.1.1 addresses parabolic- elliptic chemotaxissystems,whileTheorems 2.1.2 and 2.1.3 tackle fullyparabolicchemotaxissys- tems intwoandthreespatialdimensions,respectively.Thesefindingshavebeenpublishedin[33]. Additionally,inChapter4,weextendtheseresultstogeneralnonlinearboundaryconditionsin2D with sub-logisticsources[31]. The secondquestioninvestigatestheexistenceofagloballyboundedsolutionfor f(u) = au − uk with k ∈ (1; 2). Despitethisquestionbeingopensince2002,progresshasbeenmade. In [71], itwasdemonstratedthatlogisticsourcesarenotoptimalforpreventingblow-upsolutions. Instead, sub-logisticsourcesoftheform f(u) = au− μu2 lnp(u+e) with 0 < p< 1 ensure globalbound- edness in2Dboundeddomains.Chapter3extendsthisresulttoinvestigatesub-logisticsources in preventingblow-upphenomenafornondegenerateKeller-Segelsystems(Theorem 3.2.1) and degenerate Keller-Segelsystems(Theorem 3.2.2). Thisworkispublishedin[30]. Furthermore, Chapter 4establishesinTheorem 4.3.1 that sub-logisticsourcesalsoensuretheglobalexistence of solutionsintwo-specieschemotaxismodels[32]. Chapter5demonstratesinTheorems 5.2.1 and 5.2.2 that sub-logisticsourcespreventblow-upeveninsuperlinearsignalproductionchemo- taxis systems.Finally,blow-uppreventionbysub-logisticsourcesforchemotaxissystemswith 4 superlinear cross-diffusionratesisprovidedinTheorems 5.1.1 and 5.1.2 in Chapter5. In additiontoaddressingthesefundamentalquestions,theappendixincludesdetailedproofs for theMoseriterationtechnique,alongwithimportantinequalitiesandfundamentalresultsin regularity theory.Theseproofsarecrucialforobtaining L∞ bounds from Lp bounds with p > 1 sufficientlylargeforsolutionstovariouschemotaxissystems,withorwithoutnonlinearboundary conditions. 5 CHAPTER 2 CHEMOTAXISWITHLOGISTICSOURCESUNDERNONLINEARNEUMANN BOUNDARYCONDITION Weconsiderclassicalsolutionstothechemotaxissystemwithlogisticsource f(u) := au − u2 under nonlinearNeumannboundaryconditions ∂u ∂ν = |u|p with p > 1 in asmoothconvexbounded domain Ω ⊂ Rn where n ≥ 2. Thischapteraimstoshowthatif p < 3 2 , and  > 0, n = 2, or  is sufficientlylargewhen n ≥ 3, thentheparabolic-ellipticchemotaxissystemadmitsaunique positive global-in-timeclassicalsolutionthatisboundedin Ω × (0;∞). Thesimilarresultisalso true if p < 3 2 , n = 2, and  > 0 or p < 7 5 , n = 3, and  is sufficientlylargefortheparabolic- parabolic chemotaxissystem. 2.1 Introduction Weareconcernedinthischapterwithsolutionstothechemotaxismodelasfollows: 8>>< >>: ut = Δu − ∇ · (u∇v) + au − u2 x ∈ Ω; t ∈ (0; Tmax); vt = Δv + u − v x ∈ Ω; t ∈ (0; Tmax); (KS) in asmooth,convex,boundeddomain Ω ⊂ Rn where ; ;a;> 0,  ≥ 0, and  ∈ R. Thesystem (KS) iscomplementedwiththenonnegativeinitialconditionsin C2+γ(Ω), where ∈ (0; 1), not identically zero: u(x; 0) = u0(x); v(x; 0) = v0(x); x ∈ Ω; (2.1.1) and thenonlinearNeumannboundaryconditions @u @ = |u|p; @v @ = 0; x ∈ @Ω; t ∈ (0; Tmax): (2.1.2) where p > 1 and  is theoutwardnormalvector. The logisticterm, au−u2, introducedintheevolutionequationfor u plays aroleindescribing the growthofthepopulation.Specifically,theterm au, with a ∈ R, isthegrowthrateofpopu- lation andtheterm −u2 models additionalovercrowdingeffects.Itwasinvestigatedin[59] that the quadraticdegradationterm −u2 can preventblow-upsolutions.Infact,itwasproventhatif 6  > n−2 n  , and  = 0, thenthesolutionsexistgloballyandremainboundedatalltimeinaconvex bounded domainwithsmoothboundary Ω ⊂ Rn, where n ≥ 2. Thisresultwaslaterimprovedin [22, 25, 70] that  = n−2 n  can preventblow-upwhen  = 0. Inatwo-dimensionalspacewith  = 1, thesystem(KS) possessesauniqueclassicalsolutionwhichisnonnegativeandboundedin Ω×(0;∞) ( seee.g.[46, 47]). Theseresultswerelaterimprovedin[71, 72] byreplacingthelogis- tic sourcesbysub-logisticonessuchas au− u2 lnp(u+e) for p ∈ (0; 1). Inahigher-dimensionalspace with  = 1, thesimilarresultscanbefoundin[65] fortheparabolic-parabolicmodelwithanaddi- tional largenessassumptionof . Inadditiontotheglobalclassicalsolutions,theexistenceglobal weak solutionsresultsforanyarbitrary  > 0 were alsoobtainedin[59] fortheparabolic-elliptic models andin[28] fortheparabolic-parabolicmodelsinathree-dimensionalsystem.Furthermore, interested readersarereferredto[24, 27, 34, 35, 38, 57, 63, 74, 75] tostudymoreaboutqualitative and quantitativeworksofchemotaxissystemswithlogisticsources. The problembecomesmoreinterestingandchallengingifthehomogeneousNeumannbound- ary conditionisreplacedbythenonlinearNeumannboundarycondition.Themethodinthischapter to obtainglobalboundednessresultsisfirsttoestablisha L1 estimate, thenfor Lp0 for some p0 > 1, and finallyapplyaMoser-typeiterationtoobtainfor L∞. Althoughthisapproachhasbeenwidely applied intreatingglobalboundednessproblemsforreaction-diffusionequations([1, 2, 13]), orfor chemotaxis systems([65]), themaindifficultiesrelyheavilyontediousintegralestimations.Un- like thehomogeneousNeumannboundaryconditions,itisnotevenstraightforwardtoseewhether the totalmassofthecelldensityfunctionisgloballyboundedornotduetothenonlinearboundary term. Infact,mostofthetechnicalchallengesinthischapteraretodealwiththenonlinearbound- ary term.Fortunately,theSobolev’straceinequalityenablesustosolveapartofaproblem: Main Question: ”What isthelargestvalue p so thatlogisticdampingstillavoidsblow-up?” This typesofquestionfornonlinearparabolicequationshasbeenintensivelystudiedin1990s.To 7 be moreprecise,ifweconsider  = 0, ourproblemissimilartothefollowingPDE: 8>>>>>>< >>>>>>: Ut = ΔU − UQ x ∈ Ω; t ∈ (0; Tmax); ∂U ∂ν = UP x ∈ Ω; t ∈ (0; Tmax); U(x; 0) = U0(x) x ∈ ¯Ω: (NBC) where Ω is asmoothboundeddomainin Rn, Q; P> 1,  > 0 and U0 ∈ W1,∞(Ω) is anonneg- ative function.Thestudyconcerningtheglobalexistencewasfirstinvestigatedin[10], andthen improved in[49] for n ≥ 2. Particularly,itwasshownthat P = Q+1 2 is criticalfortheblowupin the followingsense: 1. if P < Q+1 2 then allsolutionsof(NBC) existgloballyandaregloballybounded, 2. If P > Q+1 2 (or P = Q+1 2 and  is sufficientlysmall)thenthereexistinitialfunctions U0 such thatthecorrespondingsolutionsof(NBC) blow-upin L∞−norm. In comparisontoourproblem,wehave Q = 2 and P = 3 2 is thecriticalpower.Indeed,wealso obtain thesimilarcriticalpower p = 3 2 as inTheorem 2.1.1. Noticethatthelocalexistenceof positive solutionwasnotmentionedin[10, 49, 48], anditisnotclearforustodefine UP without knowing U is nonnegative,sothepresenceofabsolutesignin(2.1.2) isnecessarytoobtainlocal positive solutionsfromnonnegative,notidenticallyzeroinitialdata. Heuristically,theanalysisdiagramcanbepresentedasfollows.Incase  = 0, bysubstituting −Δv = u − v into thefirstequationof(KS), weobtain ut = Δu + au − ∇u · ∇v + ( − )u2 − uv: If  is sufficientlylarge,thensolutionsmightbeboundedgloballysincethenonlinearterm (−)u2 might dominateothertermsincludingthenonlinearboundaryterm.Incase  = 1, wecannotsub- stitute Δv = v − u directly intothefirstequationof(KS); however,westillhavesomecertain controls of v by u from thesecondequationof(KS) thankstoSobolevinequality.Weexpectthat this intuitionshouldbetrueinlowerspacialdimensionand”weaker”nonlinearboundaryterms 8 since thecriticalSobolevexponentdecreasesifthespacialdimensionincreases.Indeed,ouranaly- sis doesnotworkfor n ≥ 4 since wedonothaveenoughroomstocontrolotherpositivenonlinear terms byusingtheterm (−)u2. Onecanalsofindsimilarideasonsub-logisticsourcepreventing 2D blow-upin[71]. Wesummarizethemainresultstoanswerapartofthemainquestion.Letusbeginwiththefol- lowing theoremfortheparabolic-ellipticcase. Theorem2.1.1. Let Ω be abounded,convexdomainwithsmoothboundaryin Rn where n ≥ 2, and  = 0. If  > n−2 n  , and 1 < p< 3 2 or  = n−2 n  with n ≥ 3 and 1 < p< 1 + 1 n then the system (KS) with initialconditions (2.1.1) and boundarycondition (2.1.2) possesses aunique positive classicalsolutionwhichremainsboundedin Ω × (0;∞). Remark 2.1.1. It isanopenquestionwhetherthereexistsaclassicalfinitetimeblow-upsolution if p ≥ 3 2 . Remark 2.1.2. The proofofborderlineboundednessinTheorem 2.1.1 when  = n−2 n  is adopted and modifiedfromtheargumentsin[22, 25, 70]. However,applyingLemma 2.3.2 to overcome challenges inboundaryintegralestimationswasnotpossible.Instead,wehadtoderiveanalter- native andimprovedestimationtohandletheboundaryterm,whichnecessitatedtheconditionof p < 1 + 1 n. The nexttheoremisfortheparabolic-parabolicsysteminatwo-dimensionalspace: Theorem2.1.2. Let Ω be abounded,convexdomainwithsmoothboundary,and  = 1, n = 2, 1 < p< 3 2 , thenthesystem (KS) with initialconditions (2.1.1) and boundarycondition (2.1.2) possesses auniquepositiveclassicalsolutionwhichremainsboundedin Ω × (0;∞). Remark 2.1.3. This theoremisanimprovementoftheresultin[73, chapter12]sincenotonlythe nonlinear boundaryconditiontakesplacebutthesmallnessassumptionofinitialdataalsoisno longer necessary. In three-dimensionalspace,weprovethefollowingtheoremfortheparabolic-paraboliccase. 9 Theorem2.1.3. Let Ω be abounded,convexdomainwithsmoothboundary,and  = 1, n = 3, 1 < p< 7 5 , thenthereexists 0 > 0 such thatforevery  >0, thesystem (KS) with initial conditions (2.1.1) and boundarycondition (2.1.2) possesses auniquepositiveclassicalsolution which remainsboundedin Ω × (0;∞). Remark 2.1.4. Hereweexpect p = 7 5 may notbethethresholdofglobalboundednessandblow-up solutions, butratherthelimitationofouranalysistools. Remark 2.1.5. Weleavetheopenquestionwhetherforevery n ≥ 4, thereexists pn > 1 such that if 1 < p 0 such that ||2 ≤ X i,j aij(x; t)ij ≤ Λ||2; (x; t) ∈ ΩT ;  ∈ Rn; (2.2.2) where aij ; bi; c ∈ Cγ(¯ΩT )(0 < < 1) and 1  ( X i,j aij Cγ(¯ΩT ) + X i bi Cγ(¯ΩT ) + ∥c∥ Cγ(¯ΩT ) ) ≤ Λγ: (2.2.3) Theorem2.2.1 ([37], p.79,Theorem4.31). Let theassumptions (2.2.2), (2.2.3) be inforce,and @Ω ∈ C2+γ(0 < < 1): Let f ∈ Cγ( ¯ ΩT ), g ∈ C1+γ(¯ΩT ) and u0 ∈ C2+γ(¯Ω) satisfying thefirst ordercompatibilitycondition: @u0 @ = g(x; 0) on @Ω: (2.2.4) Then thereexistsauniquesolution u ∈ C2+γ(¯Ω T ) to theproblem (2.2.1) with theNeumannbound- ary condition ∂u ∂ν = g on @Ω×(0; T). Moreover,thereexistsaconstant C independent of g and u0 such that ∥u∥ C2+γ(¯ΩT ) ≤ C  1  ∥f∥ Cγ(¯ΩT ) + ∥g∥ C1+γ(¯ΩT ) + ∥u0∥ C2+γ(¯Ω)  : (2.2.5) where C is dependentonlyon n; ; Λ/;Λγ and Ω. This estimate,togetherwithLeray-Schauderfixedpointargumentisthemaintoolstoprove the followingtheorem. Theorem2.2.2. If nonnegativefunctions u0; v0 arein C2+γ(¯Ω) such that @u0 @ = |u0|1+γ on @Ω; (2.2.6) where ∈ (0; 1). Thenthereexists T > 0 such thatproblem (KS) admits auniquenonnegative solution u; v in C2+γ(¯ΩT ). Moreover,if u0; v0 arenotidenticallyzeroin Ω then u; v arestrictly positive in Ω¯T . Remark 2.2.1. The convexityassumptionofdomain Ω is notnecessaryinthistheorem. Remark 2.2.2. By substituting = p − 1 into Theorem 2.2.2, weobtainlocalexistenceand uniqueness ofpositivesolutionsinTheorem 2.1.1, 2.1.2, and 2.1.3. 11 Proof. From nowtotheendofthisproof,wewilluse C as auniversalnotationforconstants differentfromtimetotime.Firstly,theshort-timeexistenceofclassicalsolutionwillbeprovedby a fixedpointargument.Let u ∈ C1+γ(¯ΩT ) be suchthat u(x; 0) = u0(x) in Ω. Then,thefunctions u0 and g(x; t) = |u(x; t)|1+γ satisfy condition(2.2.4), and g ∈ C1+γ(¯ΩT ). Weassume T < 1, and consider thesetoffunctionsgivenby BT (R) := n u ∈ C1+γ(¯ΩT ) such that ∥u∥ C1+γ(¯ΩT ) ≤ R o : Now wedefinethemap A : BT (R) −→ C1+γ(¯ΩT ) where Au := U is asolutionof 8>>< >>: Ut = ΔU − ∇ · (u∇V ) + au − u2 x ∈ Ω; t ∈ (0; Tmax); Vt = ΔV + u − V x ∈ Ω; t ∈ (0; Tmax); (2.2.7) under Neumannboundarycondition: @U @ = |u|1+γ; @V @ = 0; x ∈ @Ω; t ∈ (0; Tmax); (2.2.8) and initialdata (U(x; 0); V (x; 0)) =(u0(x); v0(x)) in Ω. Wefirstprovethat A sends bounded sets intorelativecompactsetsof C1+γ(¯ΩT ). Indeed,theinequality(2.2.5) impliesthereexists R′ > 0 independent of T such that ∥Au∥ C2+γ(¯ΩT ) ≤ R′ for all u in BT (R). Asboundedsetsin C2+γ(¯ΩT ) are relativelycompactin C1+γ(¯ΩT ). Weclaimthat A is continuous.Infact,let un → u in C1+γ(¯ΩT ), weneedtoprove Un := Aun → U := Au in C1+γ(¯ΩT ). Nowwecanseethat Un−U satisfies 8>>< >>: (Un − U)t = Δ(Un − U) + fn; x ∈ Ω; t ∈ (0; Tmax);  (Vn − V )t = Δ(Vn − V ) + (un − u) − (Vn − V ) x ∈ Ω; t ∈ (0; Tmax); (2.2.9) where fn := −∇· (un∇Vn −u∇V )+un(a−un)−u(a−u). Onecanverifythat fn satisfies the assumptionsofTheorem 2.2.1. Plus,theboundarycondition @(Un − U) @ = |un|1+γ − |u|1+γ; @(Vn − V ) @ = 0; x ∈ @Ω; t ∈ (0; Tmax): (2.2.10) 12 We claim that Vn → V in C2+γ(Ω¯T ) for  ≥ 0. Indeed,when  > 0, wemakeuseof(2.2.5) and when  = 0, weapplySchaudertypeestimateforellipticequationtoobtainthat Vn → V in C2+γ(Ω¯T ). Thisleadsto fn → 0 in Cγ(¯ΩT ), combinewithinequality(2.2.5) entailthat Un → U in C2+γ(¯ΩT ). InordertoapplytheLeray-Schauderfixedpointtheoremwejusthavetoprovethat if T is sufficientlysmall,and R ≥ 2(1+d(Ω)1−γ) ∥u0∥ C2+γ(¯Ω), then A(BT (R)) ⊂ BT (R). Indeed, |Au(x; t)| ≤|Au(x; 0)| + t ∥DtAu∥ C0(¯Ω) ≤ ∥u0∥ C0(¯Ω) + TR′ |Au(x; t) − Au(x; s)| |t − s| 1+γ 2 ≤ ∥DtAu∥ C0(¯ΩT ) |t − s| 1−γ 2 ≤ R′T 1−γ 2 ; and, |DxAu(x; t) − DxAu(y;s)| |x − y|γ + |t − s| γ 2 ≤ |t − s|1−γ 2 D2x Au C0(¯ΩT ) + |x − y|1−γ(s γ 2 + D2u0 C0(¯Ω)) ≤ T1−γ 2R′ + d(Ω)1−γT γ 2R′ + d(Ω)1−γ D2u0 C0(¯Ω) : These aboveestimatesimplythat ∥Au∥ C1+γ(¯ΩT ) ≤ R 2 + R′T + R′T 1−γ 2 + T1−γ 2R′ + d(Ω)1−γT γ 2R′: Since R′ is independentof T for all T < 1, wecanchoose T sufficientlysmallastohave R′T + R′T 1−γ 2 + T1−γ 2R′ + d(Ω)1−γT γ 2R′ ≤ R 2 : This furtherimpliesthat ∥Au∥ C1+γ(¯ΩT ) ≤ R for all u ∈ BT (R): Thus A has afixedpointin BT (R). Nowif u is afixedpointof A, u ∈ C2+γ(¯ΩT ) and itisa solution of(KS). Secondly,thenonnegativityofsolutionswillbeprovedbythetruncationmethod:Letting  := min {u; 0} and (t) := 1 2 R Ω 2 dx, weseethat is continuouslydifferentiablewiththederivative ′(t) = − Z Ω |∇|2 + a Z Ω 2 + Z ∂Ω |u|1+γ dS +  Z Ω ∇ · ∇v −  Z Ω 3 ≤ − Z Ω |∇|2 + a Z Ω 2 +  Z Ω ∇ · ∇v −  Z Ω 3: (2.2.11) 13 WemakeuseofYoung’sinequalitycombinedwiththeglobalboundednessof |∇v| in ¯ΩT to obtain  Z Ω ∇ · ∇v ≤  Z Ω |∇|2 + C Z Ω 2; (2.2.12) for some C > 0. Wealsohave − R Ω 3 ≤ C R Ω 2, where C =  supΩT |u(x; t)|. Thistogether with (2.2.11), (2.2.12) impliesthat ′(t) ≤ C (t) for all 0 < t @u @ (x0; t0) = |u(x0; t0)|1+γ = 0: Thus, u > 0 and bysimilarargumentswealsohave v > 0. Finally,theuniquenessofclassicalsolutionswillbeprovedbyacontradictionproof.Assuming (u1; v1 and (u2; v2) are twopositiveclassicalsolutionsofthesystem(KS). Let U := u1 − u2, V := v1 − v2, then (U;V ) is asolutionofthefollowingsystem: 8>>< >>: Ut = ΔU + F; x ∈ Ω; t ∈ (0; Tmax); Vt = ΔV + U − Vx ∈ Ω; t ∈ (0; Tmax); (2.2.13) where F := −∇(u1∇v1 − u2∇v2) + f(u1) − f(u2), andtheboundarycondition @U @ = |u1|1+γ − |u2|1+γ; @V @ = 0; x ∈ @Ω; t ∈ (0; Tmax): (2.2.14) By meanvaluetheorem,thereexists z(x; t) between u1(x; t) and u2(x; t) such that u1(x; t) − u2(x; t) =(u1(x; t) − u2(x; t))f′(z(x; t)): Multiplying thefirstequationof(2.2.13) by U implies 1 2 d dt Z Ω U2 dx = − Z Ω |∇U|2 + Z ∂Ω U(|u1|1+γ − |u2|1+γ) dS +  Z Ω (u1∇v1 − u2∇v2) · ∇U + Z Ω U2f′(z): (2.2.15) 14 Wemakeuseoftheglobalboundednesspropertyof u1; u2 in ¯ΩT , thereafterapplySobolev’strace theorem, andfinallyYoung’sinequalitytohave Z ∂Ω U(|u1|1+γ − |u2|1+γ) dS ≤ C Z ∂Ω U2 dS ≤  Z Ω |∇U|2 + C() Z Ω U2: (2.2.16) Since, u1∇v1 − u2∇v2 = U∇v1 + u2∇V; we have  Z Ω ∇U · (u1∇v1 − u2∇v2) ≤ C Z Ω U|∇U| + |∇U||∇V | ≤  Z Ω |∇U|2 + C Z Ω U2 + C Z Ω |∇V |2: (2.2.17) Wealsohave R Ω U2f′(z) ≤ C R Ω U2 where C = supmin {u1,u2}≤z≤max {u1,u2} |f′(z)|. Multiplying the secondequationof(2.2.13) by V , andapplyingYoung’sinequality,weobtain d dt Z Ω V 2 + Z Ω |∇V |2 ≤ C Z Ω U2: (2.2.18) From (2.2.15) to(2.2.18), weobtain d dt Z Ω U2 + Z Ω V 2  ≤ C Z Ω U2 + Z Ω V 2  : (2.2.19) The initialconditionsandGronwall’sinequalityimplythat U ≡ V ≡ 0, andthusthereisaunique solution tothesystem(KS). 2.3 Preliminaries The nextlemmagivinganusefulestimatewilllaterbeappliedinSection 2.4. Interestedreaders are referredto[56, 72] formoredetailsabouttheproof. Lemma 2.3.1. Let Ω ⊂ R2 be aboundeddomainwithsmoothboundary,andlet p > 1 and 1 ≤ r 0 such thatforeach  > 0, onecanpick C() > 0 such that ∥u∥p Lp(Ω) ≤  ∥∇u∥p−r L2(Ω) ∥u ln |u|∥r Lr(Ω) + C ∥u∥p Lr(Ω) + C() (2.3.1) holds forall u ∈ W1,2(Ω). 15 The followinglemmaprovidingestimatesontheboundarywillbeusefulinSection 2.4. Lemma 2.3.2. If r ≥ 1, p ∈ (1; 3 2 ), and g ∈ C1(¯Ω); then forevery  > 0, thereexistsaconstant C = C(;Ω; p;r) such that Z ∂Ω |g|p+2r−1 ≤  Z Ω |g|2r+1 +  Z Ω |∇gr|2 + C: (2.3.2) Proof. Let  := |g|r, wehave 2+p−1 r ∈ W1,1(Ω). Tracetheorem, W1,1(Ω) → L1(@Ω), yields Z ∂Ω 2+p−1 r ≤ c1 Z Ω 2+p−1 r + (2+ p − 1 r )c1 Z Ω 1+p−1 r |∇| ≤ c1 Z Ω 2+p−1 r + 3c1 Z Ω 1+p−1 r |∇| (2.3.3) where c1 = c1(n; Ω) > 0. ByYoung’sinequality,thefollowingholdsforall  > 0 3c1 Z Ω 1+p−1 r |∇| ≤  Z Ω |∇|2 + c21  Z Ω 2+2(p−1) r : (2.3.4) WeapplyYoung’sinequalityagaintoobtainafurtherestimate c1 Z Ω 2+p−1 r + c21  Z Ω 2+2(p−1) r ≤  Z Ω 2+1 r + c2: (2.3.5) where c2 depending on ; p;r;n;Ω. Wecompletetheproofof(2.3.2) bycollecting(2.3.3),(2.3.4) and (2.3.5) together. The followinglemmaisanessentialestimatetoobtain L2 bounds from L ln L bounds. Lemma 2.3.3. If p ∈ (1; 7 5 ), n = 3, and (u; v) arein C1(¯Ω × (0; Tmax)) and Z Ω |∇v(·; t)|2 ≤ A (2.3.6) holds forall t ∈ (0; Tmax), thenforevery  > 0, thereexistsaconstant C = C(;Ω; p;A) such that Z ∂Ω up|∇v|2 ≤  Z Ω  u3 + |∇u|2 + u2|∇v|2 + ∇|∇v|2 2  + C: (2.3.7) holds forall t ∈ (0; Tmax). 16 Proof. By tracetheorem W1,1(Ω) → L1(@Ω), Z ∂Ω |u|p|∇v|2 ≤ c1 Z Ω |u|p|∇v|2 + c1 Z Ω |u|p ∇|∇v|2 + c1p Z Ω |u|p−1|∇u||∇v|2 (2.3.8) where c1 = c1(Ω) > 0. ApplyYoung’sinequalityyields c1 Z Ω |u|p|∇v|2 ≤  2 Z Ω u2|∇v|2 + 2 − p 2  p c1 2−p p Z Ω |∇v|2 ≤  2 Z Ω u2|∇v|2 + c2 (2.3.9) where c2 := 2−p 2 A  p ϵc1 2−p p . Notethat 2p < 3, weapplyYoung’sinequalitytoobtain c1 Z Ω |u|p ∇|∇v|2 ≤  2 Z Ω ∇|∇v|2 2 + 1 2 Z Ω u2p ≤  2 Z Ω ∇|∇v|2 2 +  Z Ω u3 + c3 (2.3.10) where c3 = c3(;Ω; p). ByYoung’sinequality, c1p Z Ω |u|p−1|∇u||∇v|2 ≤  2 Z Ω u2|∇v|2 + c4 Z Ω |∇u| 2 3−p |∇v|2 ≤  2 Z Ω u2|∇v|2 +  Z Ω |∇u|2 + c5 Z Ω |∇v| 6−2p 2−p (2.3.11) where c4; c5 are positiveanddependenton ;Ω; p. Here,weusethecondition 1 < p< 7 5 to obtain 3−p 2−p < 8 3 . ByYoung’sinequality, c5 Z Ω |∇v| 6−2p 2−p ≤  Z Ω |∇v| 16 3 + c6 (2.3.12) where c6 = c6(;;p; Ω). InlightofGagliardo-Nirenberginequality, |∇v|2 L 83 (Ω) ≤ cGN ∇|∇v|2 3 4 L2(Ω) |∇v|2 1 4 L1(Ω) + cGN |∇v|2 L1(Ω) ≤ cGNA 1 4 ∇|∇v|2 3 4 L2(Ω) + cGNA: (2.3.13) Hence  Z Ω |∇v| 16 3 ≤   cGNA 1 4 ∇|∇v|2 3 4 L2(Ω) + cGNA 8 3 ≤ 25/3(cGNA 1 4 ) 8 3  Z Ω ∇|∇v|2 2 + 25/3(cGNA) 8 3 : (2.3.14) 17 Choosing  such that 25/3(cGNA 1 4 ) 8 3  = , andplugginginto(2.3.14), (2.3.12), and(2.3.11) re- spectively,weobtain c1p Z Ω |u|p−1|∇u||∇v|2 ≤  Z Ω ∇|∇v|2 2 +  2 Z Ω u2|∇v|2 +  Z Ω |∇u|2 + c7 (2.3.15) where c7 = c6 + 25/3(cGNA) 8 3 . Wefinallycompletetheproofof(2.3.7) bysubstituting(2.3.9), (2.3.10) and(2.3.15) into(2.3.8). 2.4 Aprioriestimates Let usfirstgiveaprioriestimatefortheparabolic-ellipticsystem, Lemma 2.4.1. If  > 0 and p ∈ (1; 3 2 ), forall r ∈ (1; χα (χα−μ)+ ) then thereexists c = c(r; ∥u0∥ Lr(Ω)) > 0 such that ∥u(·; t)∥ Lr(Ω) ≤ c; ∀t ∈ (0; Tmax): (2.4.1) Proof. Multiplying thefirstequationinthesystem(KS) by u2r−1 yields 1 2r d dt Z Ω u2r = Z Ω u2r−1ut = Z Ω u2r−1 [Δu − ∇(u∇v) + f(u)] = −2r − 1 r2 Z Ω |∇ur|2 −  2r − 1 2r Z Ω u2rΔv + Z Ω f(u)u2r−1 + Z ∂Ω u2r+p−1 dS = −2r − 1 r2 Z Ω |∇ur|2 + 2r − 1 2r Z Ω u2r( u −  v) + Z ∂Ω u2r+p−1 dS + a Z Ω u2r −  Z Ω u2r+1: (2.4.2) Since v ≥ 0, wehave d dt Z Ω u2r ≤ −2(2r − 1) r Z Ω |∇ur|2 − [2r −  (2r − 1)] Z Ω u2r+1 + 2r Z ∂Ω u2r+p−1 dS + 2ra Z Ω u2r: (2.4.3) By Lemma 2.3.2, weobtain 2r Z ∂Ω u2r+p−1 dS ≤ 2r Z Ω |∇ur|2 + 2r Z Ω u2r+1 + c2: (2.4.4) 18 WemakeuseofYoung’sinequalitytoobtain (2ra + 1) Z Ω u2r ≤  Z Ω u2r+1 + c3: (2.4.5) Collecting (2.4.3), (2.4.4) and(2.4.5), wehave d dt Z Ω u2r + Z Ω u2r ≤  2r − 2(2r − 1) r  Z Ω |∇ur|2 − [2r −  (2r − 1) − 2] Z Ω u2r+1 + c4: (2.4.6) If χα (χα−μ)+ > 2r > 1, thenselecting  = min n 2r−1 r2 ; 2rμ−χα(2r−1) 2 o and plugginginto(2.4.6), we deduce d dt Z Ω u2r + Z Ω u2r ≤ c2 (2.4.7) This yields(2.4.1), hencetheproofiscomplete. In theparabolic-paraboliccase  = 1, thefollowinglemmagivesusaprioriboundsforsolution of (KS) withinitialdata(2.1.1) andtheboundarycondition(2.1.2). Lemma 2.4.2. If 1 < p< 3 2 , and (u; v) is aclassicalsolutionto (KS) with initialdata (2.1.1) and the boundarycondition (2.1.2) without theconvexityassumptionof Ω, and n ≥ 2 then thereexists a positiveconstant C such that Z Ω (u(·; t) +1) ln (u(·; t) +1)+ Z Ω |∇v(·; t)|2 ≤ C (2.4.8) for all t ∈ (0; Tmax). Proof. Let denote y(t) := R Ω(u(·; t) +1) ln (u(·; t) +1)+ R Ω |∇v(·; t)|2, wehave y′(t) = Z Ω [Δu − ∇ · (u∇v) + f(u)] [ln (u + 1)+1] + 2 Z Ω ∇v · ∇(Δv + u − v) := I1 + I2: (2.4.9) 19 By integrationbyparts, I1 can berewrittenas I1 = − Z Ω |∇u|2 u + 1 +  Z Ω u u + 1 ∇u · ∇v + a Z Ω u [ln (u + 1)+1] −  Z Ω u2 [ln (u + 1)+1]+ Z ∂Ω up [ln (u + 1)+1] dS (2.4.10) By integrationbyparts,Cauchy-Schwarzinequalityandelementaryinequalityln(u + 1) ≤ u, we have  Z Ω u u + 1 ∇u · ∇v =  Z Ω ∇(u − ln(u + 1)) · ∇v = − Z Ω (u − ln(u + 1))Δv ≤ 1 2 Z Ω (Δv)2 + 2 Z Ω u2: (2.4.11) One canverifythatthereexists c1(; a) > 0 satisfying u [ln (u + 1)+1] ≤  4a u2 [ln (u + 1)+1]+ c1; hence, a Z Ω u [ln (u + 1)+1] ≤  4 Z Ω u2 [ln (u + 1)+1]+ c1: (2.4.12) In lightofSobolev’stracetheorem, W1,1(Ω) ,→ L1(@Ω), thereexists c2(Ω) > 0 such that Z ∂Ω up [ln (u + 1)+1] dS ≤ c2 Z Ω up [ln (u + 1)+1]+ pc2 Z Ω up−1|∇u| [ln (u + 1)+1] + c2 Z Ω up u + 1 |∇u| [ln (u + 1)+1] : (2.4.13) By Young’sinequality,wehave pc2 Z Ω up−1|∇u| [ln (u + 1)+1] ≤ 1 4 Z Ω |∇u|2 u + 1 + pc2 Z Ω u2p−2(u + 1)[ln (u + 1)+1]2 ; (2.4.14) and c2 Z Ω up u + 1 |∇u| [ln (u + 1)+1] ≤ 1 4 Z Ω |∇u|2 u + 1 + c22 Z Ω u2p u + 1 [ln (u + 1)+1]2 : (2.4.15) 20 By thesimilarargumentasin(2.4.12), thereexists c3(p;Ω; ) > 0 such that c2 Z Ω up [ln (u + 1)+1]+ pc2 Z Ω u2p−2(u + 1)[ln (u + 1)+1]2 + c22 Z Ω u2p u + 1 [ln (u + 1)+1]2 ≤  4 Z Ω u2 [ln (u + 1)+1]+ c3: (2.4.16) From (2.4.13) to(2.4.16), weobtain Z ∂Ω up [ln (u + 1)+1] dS ≤ 1 2 Z Ω |∇u|2 u + 1 +  4 Z Ω u2 [ln (u + 1)+1]+ c3: (2.4.17) Now,wehandle I2 as follows: I2 = −2 Z Ω (Δv)2 − 2 Z Ω |∇v|2 + 2 Z Ω ∇u · ∇v: (2.4.18) By integrationbypartandYoung’sinequality,wehave 2 Z Ω ∇u · ∇v ≤ 1 2 Z Ω (Δv)2 + 2 2 Z Ω u2: (2.4.19) One canverifythatthereexists c4( ; ;; Ω) > 0 such that (2 + 2 2) Z Ω u2 + 2 Z Ω (u + 1) ln (u + 1) ≤  4 Z Ω u2 [ln (u + 1)+1]+ c4: (2.4.20) Collecting (2.4.10), (2.4.12), (2.4.17) andfrom(2.4.18) to(2.4.20), weobtain y′(t)+2 y(t) ≤ −1 2 Z Ω |∇u|2 u + 1 − 4 Z Ω u2 [ln (u + 1)+1]+c5 ≤ c5; ∀t ∈ (0; Tmax); (2.4.21) where c5 = 2 and c5 := c1 + c3 + c4. This,togetherwiththeGronwall’sinequality,yields y(t) ≤ e−2βty(0) + c5 2 (1 − e−2βt) ≤ C where C := max n y(0); c5 2β o , andtheproofof(2.4.8) iscomplete. The followinglemmagivesan L2-bound intwo-dimensionalspacefortheparabolic-parabolic system. 21 Lemma 2.4.3. If  = 1, n = 2, 1 < p< 3 2 , and (u; v) is aclassicalsolutionto (KS) with initial data (2.1.1) and theboundaryconditionthenthereexistsapositiveconstant C such that Z Ω u2(·; t) + Z Ω |∇v(·; t)|4 ≤ C (2.4.22) for all t ∈ (0; Tmax). Proof. Let denote (t) := 1 2 Z Ω u2 + 1 4 Z Ω |∇v|4; we have ′(t) = Z Ω u [Δu − ∇ · (u∇v) + f(u)] + Z Ω |∇v|2∇v · ∇(Δv + u − v) := J1 + J2: (2.4.23) By integrationbyparts,weobtain J1 = − Z Ω |∇u|2 +  Z Ω u∇u · ∇v + a Z Ω u2 −  Z Ω u3 + Z ∂Ω up+1 dS: (2.4.24) By Young’sinequality,wehave  Z Ω u∇u · ∇v + a Z Ω u2 ≤ 1 2 Z Ω |∇u|2 + 2 Z Ω u2|∇v|2: (2.4.25) In lightofSobolev’stracetheorem, W1,1(Ω) ,→ L1(@Ω), thereexists c1 := c1(Ω) > 0 such that Z ∂Ω up+1 dS ≤ c1 Z Ω up+1 + c1(p + 1) Z Ω up|∇u|: (2.4.26) Since 1 < p< 3 2 , weapplyYoung’sinequalitytoobtain Z ∂Ω up+1 dS ≤  4 Z Ω u3 + 1 4 Z Ω |∇u|2 + c1(p + 1)2 Z Ω u2p + c2 ≤ 1 4 Z Ω |∇u|2 +  2 Z Ω u3 + c3; (2.4.27) 22 where c2; c3 > 0 depending onlyon p; ;Ω. Todealwith J2, wemakeuseofthefollowingpoint- wise identity ∇v · ∇Δv = 1 2 Δ(|∇v|2) − |D2v|2 to obtain J2 = −1 2 Z Ω |∇|∇v|2|2 − Z Ω |∇v|2|D2v|2 + Z Ω |∇v|2∇v · ∇u − Z Ω |∇v|4 + 1 2 Z ∂Ω @|∇v|2 @ |∇v|2: (2.4.28) Applying Lemma B.0.6 and thepointwiseinequality (Δv)2 ≤ 2|D2v|2 to (2.4.28), wededuce J2 ≤ −1 2 Z Ω |∇|∇v|2|2 − Z Ω |∇v|4 − 1 2 Z Ω |∇v|2|Δv|2 + Z Ω |∇v|2∇v · ∇u: (2.4.29) By integralbypartsandYoung’sinequality,weobtain Z Ω |∇v|2∇v · ∇u = − Z Ω u∇|∇v|2 · ∇v − Z Ω u|∇v|2Δv ≤ 1 4 Z Ω |∇|∇v|2|2 + 1 4 Z Ω |∇v|2|Δv|2 + 2 2 Z Ω u2|∇v|2 (2.4.30) Collecting from(2.4.23) to(2.4.30) yields ′ + 4  ≤ −1 4 Z Ω |∇u|2 − 1 2 Z Ω |∇|∇v|2|2 −  2 Z Ω u3 + c4 Z Ω u2|∇v|2 + c5 Z Ω u2 + c3 (2.4.31) where c3; c4; c5 are positiveconstantsdependingon ; ;;a. ByYoung’sinequality c4 Z Ω u2|∇v|2 ≤ c4 Z Ω |∇v|6 + √c4  Z Ω u3 (2.4.32) By Gagliardo-Nirenberginequalityfor n = 2 and (2.4.8), thereexists cGN > 0 such that Z Ω |∇v|6 ≤ cGN Z Ω |∇|∇v|2|2 Z Ω |∇v|2  + cGN Z Ω |∇v|2 3 ≤ c6 Z Ω |∇|∇v|2|2  + c7; (2.4.33) 23 where c6; c7 are positiveconstantsdependingon cGN and supt∈(0,Tmax) R Ω |∇v|2. Wemakeuseof (2.3.1) for n = 2 and (2.4.8) toobtain Z Ω u3 ≤  Z Ω |∇u|2 Z Ω u| ln u|  + C Z Ω u 3 + c() ≤ c8 Z Ω |∇u|2 + c9; (2.4.34) where c8 := supt∈(0,Tmax) R Ω u| ln u| and c9 > 0 depending on  and supt∈(0,Tmax) R Ω u. Inlightof Young’sinequality c5 Z Ω u2 ≤  4 Z Ω u3 + c10: (2.4.35) where c10 > 0 depending on c5; ; |Ω|. Combiningfrom(2.4.31) to(2.4.35), wehave ′ + 4  ≤  c4c8 √  − 1 4 Z Ω |∇u|2 +  c4c6 − 1 4 Z Ω |∇|∇v|2|2 + c11; (2.4.36) where c11 > 0 depending on . Choosing  sufficientlysmallandsubstitutinginto(2.4.36), we obtain ′ + 4  ≤ c11: (2.4.37) This, togetherwithGronwall’sinequalityyields (t) ≤ C := max n (0); c11 4β o for all t ∈ (0; Tmax), and theproofofLemma 2.4.3 is complete. The nextlemmaisthekeystepintheproofoftheparabolic-parabolicsysteminthree-dimensio- nal space. Lemma 2.4.4. Let (u; v) be aclassicalsolutionto (KS) in aconvexboundeddomain Ω with smooth boundary.If  = 1, n = 3, 1 < p< 7 5 and  is sufficientlylarge,thenthereexistsapositive constant C such that Z Ω u2(·; t) + Z Ω |∇v(·; t)|4 ≤ C (2.4.38) for all t ∈ (0; Tmax). Remark 2.4.1. When welookattheproofofTheorem 2.1.2 carefully, n = 2 is utilizedtoestimate R Ω u2|∇v|2. For n = 3, inordertoeliminatethistermweborrowtheideaasin[53, 69] by 24 introducinganextraterm R Ω u|∇v|2, inthefunction  for Theorem 2.1.2. Notethattheterm ut|∇v|2 will introduce −u2|∇v|2, andasufficientlylargeparameter  will helptheestimates. Proof. Let call (t) := R Ω u2 + R Ω |∇v|4 + 1 3 R Ω u|∇v|2, wehave ′(t) =2 Z Ω u [Δu − ∇ · (u∇v) + f(u)] + 4 Z Ω |∇v|2∇v · ∇(Δv + u − v) + 2 3 Z Ω u∇v · ∇(Δv + u − v) + 2 3 Z Ω |∇v|2 [Δu − ∇ · (u∇v) + f(u)] := K1 + K2 + K3 + K4: (2.4.39) By integrationbyparts, K1 can bewrittenas: K1 = −2 Z Ω |∇u|2 + 2 Z ∂Ω up+1 dS + 2 Z Ω u∇u · ∇v + 2a Z Ω u2 − 2 Z Ω u3: (2.4.40) By Lemma 2.3.2, weobtain 2 Z ∂Ω up+1 dS ≤ 21 Z Ω u3 + 21 Z Ω |∇u|2 + 2c1: (2.4.41) By Young’sinequality,wehave 2 Z Ω u∇u · ∇v ≤ 1 Z Ω |∇u|2 +  1 Z Ω u2|∇v|2: (2.4.42) Wealsohave 2a Z Ω u2 ≤ 2a1 Z Ω u3 + c2; (2.4.43) where c2 > 0 depending on 1. From(2.4.40) to(2.4.43), weobtain K1 ≤ [(2 + )1 − 2] Z Ω |∇u|2 + 2[(a + 1)1 − ] Z Ω u3 +  1 Z Ω u2|∇v|2 + c3; (2.4.44) 25 where c3 = 2+ c2. Wechoose 1 = 1 2 min n 2 2+χ; μ a+1 o and substituteinto(2.4.44) toobtain K1 ≤ − Z Ω |∇u|2 −  Z Ω u3 +  1 Z Ω u2|∇v|2 + c3: (2.4.45) Todealwith K2, weusesimilarestimatesto(2.4.28) and(2.4.29) inestimating J2 in Lemma 2.4.3 to obtain K2 + 4 Z Ω |∇v|4 ≤ 2( 2 − 1) Z Ω |∇|∇v|2|2 +  2 2 + 3 2 Z Ω u2|∇v|2: (2.4.46) By substituting 2 = 1 2α into (2.4.46), wededuce K2 + 4 Z Ω |∇v|4 ≤ − Z Ω |∇|∇v|2|2 + 7 2 Z Ω u2|∇v|2 (2.4.47) Todealwith K3, wemakeuseofthefollowingidentity ∇v · ∇Δv = 1 2 Δ(|∇v|2) − |D2v|2: Rewriting K3 as K3 = −1 3 Z Ω ∇u · ∇|∇v|2 − 2 3 Z Ω u|D2v|2 + 2 3 Z Ω u∇v · ∇u − 2 3 Z Ω u|∇v|2 + 1 3 Z ∂Ω u @|∇v|2 @ : (2.4.48) WedropthelasttermduetoLemma B.0.6, neglectthesecondtermandapplyCauchy-Schwartz inequality ab ≤ a2 + 1 4 b2; to thethirdandtheforthtermswithasufficientlysmall  to obtain K3 + 2 3 Z Ω u|∇v|2 ≤ 1 3 Z Ω |∇|∇v|2|2 + 2 8 Z Ω u3 + 1 3 Z Ω |∇u|2: (2.4.49) By integrationbyparts, K4 can berewrittenas: K4 = −1 3 Z Ω ∇|∇v|2 · ∇u + 1 3 Z ∂Ω up|∇v|2 +  3 Z Ω ∇|∇v|2 · u∇v + a 3 Z Ω u|∇v|2 −  3 Z Ω u2|∇v|2: (2.4.50) 26 By Cauchy-Schwarzinequality,weobtain −1 3 Z Ω ∇|∇v|2 · ∇u ≤ 1 6 Z Ω ∇|∇v|2 2 + 1 3 Z Ω |∇u|2: (2.4.51) In lightofLemma 2.3.3 and Lemma 2.4.2, thefollowinginequality 1 3 Z ∂Ω up|∇v|2 ≤ 1 3 Z Ω |∇u|2 + u2|∇v|2 + ∇|∇v|2 2 + u3 + c1 (2.4.52) holds forall t ∈ (0; Tmax), withsomepositiveconstant c1. ByYoung’sinequality,weobtain  3 Z Ω ∇|∇v|2 · u∇v ≤ 1 6 Z Ω ∇|∇v|2 2 + 2 3 Z Ω u2|∇v|2; (2.4.53) and a 3 Z Ω u|∇v|2 ≤ 1 3 Z Ω u2|∇v|2 + c2; (2.4.54) with somepositiveconstant c2. Combiningfrom(2.4.50) to(2.4.54), weobtain K4 ≤ 2 3 Z Ω |∇u|2 + ∇|∇v|2 2 + 2 −  + 2 3 Z Ω u2|∇v|2 + 1 3 Z Ω u3 + c3; (2.4.55) with somepositiveconstant c3. Collecting(2.4.45), (2.4.47), (2.4.49), and(2.4.55), wehave ′ + 4 Z Ω |∇v|4 + 2 3 Z Ω u|∇v|2 ≤  7 2 +  1 + 2 −  + 2 3 Z Ω u2|∇v|2 +  1 3 −  Z Ω u3 + 2 8 Z Ω u2 + c5: (2.4.56) This leadsto ′ + 2 ≤  7 2 +  1 + 2 −  + 2 3 Z Ω u2|∇v|2 +  1 3 −  Z Ω u3 + 16 + 2 8 Z Ω u2 + c5: (2.4.57) By Young’sinequality,forevery  > 0, thereexistsapositiveconstant c6 = c6() such that: 16 + 2 8 Z Ω u2 ≤  Z Ω u3 + c6: (2.4.58) 27 Therefore, weneedtochoose 0 sufficientlylargesuchthat 8>>>>>>< >>>>>>: 7 2 + χ ϵ1 + χ2−μ0+2 3 ≤ 0 1 3 +  − 0 ≤ 0 0 ≥ 2(a+1) 2+χ : Therefore, if  >0, where 0 := max  1 3 ; 2(a + 1) 2 +  ; 3   2 +  + 7 2 + 2 + 2 2  (2.4.59) and then(2.4.56) yields ′ + 2 ≤ c6: Applying Gronwall’sinequality,weseethat (t) ≤ max  (0); c6 2  (2.4.60) and therebyconcludetheproof. Remark 2.4.2. 0 defined asin (2.4.56) is notsharp.Weleavetheopenquestiontoobtainan optimal formula 0. 2.5 Globalboundedness In thissection,weshowthatif u is uniformlyboundedintimeunder ∥·∥ Lr0(Ω), thenitisalso uniformly boundedintimeunder ∥·∥ L∞(Ω). Theorem2.5.1. Let r0 > n 2 and (u; v) be aclassicalsolutionof (KS) on Ω×(0; Tmax) with maximal existence time Tmax ∈ (0;∞]. If sup t∈(0,Tmax) ∥u(·; t)∥ Lr0(Ω) < ∞; then sup t∈(0,Tmax)  ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω)  < ∞: 28 Proof. It isadirectconsequenceofTheorem C.2.1 for f(u) = au − u2 and g(u) = uq with q ∈ 􀀀 1; 3 2  . Wearenowreadytoproveourmainresults.Letusbeginwiththeproofoftheparabolic-elliptic system. ProofofTheorem 2.1.1. Throughout thisproof,unlessspecifiedotherwise,thenotation C represents constantsthatmayvaryfromtimetotime.Incase  > n−2 n  , weobtain χα (χα−μ)+ > n 2 . We first applyLemma 2.4.1 to have u ∈ L∞ ((0; Tmax); Lq(Ω)) for any q ∈  n 2 ; χα (χα−μ)+  and thereby conclude that u ∈ L∞ ((0; Tmax); L∞(Ω)) thank toTheorem 2.5.1. When  = n−2 n  , let w = u n 4 , andapplytraceembeddingTheorem W1,1(Ω) → L1(@Ω), we have Z ∂Ω w2+4(p−1) n ≤ C Z Ω w2+4(p−1) n + C  2 + 4(p − 1) n Z Ω w1+4(p−1) n |∇w| ≤ C Z Ω w2+4(p−1) n + 3C Z Ω w1+4(p−1) n |∇w| ≤ C Z Ω w2+4(p−1) n +  2 Z Ω |∇w|2 + C() Z Ω w2+8(p−1) n ; (2.5.1) where thelastinequalitycomesfromYoung’sinequalityforanyarbitrary  > 0. ByLemma B.0.2, we have Z Ω w2+8(p−1) n ≤ C Z Ω |∇w|2 ¯pa 2 Z Ω w 4 n n¯p(1−a) 4 + C Z Ω w 4 n n¯p 4 ; (2.5.2) where ¯p = 2+ 8(p − 1) n ; a = n2(n + 4(p − 1) +2) (n + 4(p − 1))(n2 − 2n + 4) ; ¯pa 2 = n2 + 4(p − 1)n − 2n n2 − 2n + 4 < 1; since p < 1 + 1 n : Wemakeuseofuniformlyboundednessof R Ω u and thenapplyYoung’sinequalityinto(2.5.2) to obtain: Z Ω w2+8(p−1) n ≤  2 Z Ω |∇w|2 + C(); (2.5.3) 29 for any  > 0. WeapplyYoung’sinequalityin(2.5.1) andthenuse(2.5.2) tohave Z ∂Ω w2+4(p−1) n ≤  Z Ω |∇w|2 + C(): (2.5.4) This impliesthat Z ∂Ω u n 2 +p−1 ≤  Z Ω |∇u n 4 |2 + C(): (2.5.5) Substitute r = n 4 and  = n−2 n  into (2.4.3), wehave d dt Z Ω u n 2 ≤ −4(n − 2) n Z Ω |∇u n 4 |2 + n 2 Z ∂Ω u n 2 +p−1 dS + na 2 Z Ω u n 2 : (2.5.6) WeapplyLemma B.0.2 and uniformboundednessof R Ω u to obtainthat Z Ω u n 2 ≤  Z Ω |∇u n 4 |2 + C(): (2.5.7) This, togetherwith(2.5.5) and(2.5.6) with  sufficientlysmallimpliesthat d dt Z Ω u n 2 + Z Ω u n 2 ≤ C: Thus, byGronwall’sinequalityweobtainthat sup t∈(0,Tmax) Z Ω u n 2 < ∞: (2.5.8) For any  ∈ (0; 1), wechoose n 4 < r< n 4 + nϵ 4χα and substituteinto(2.4.6) toobtainthat d dt Z Ω u2r + Z Ω u2r ≤  2r − 2(2r − 1) r  Z Ω |∇ur|2 + 3 Z Ω u2r+1 + C: (2.5.9) Setting z := ur and applyinginterpolationinequality,wehave Z Ω z2+1 r ≤ Z Ω z 2n n−2 n−2 n Z Ω z n 2r 2 n : (2.5.10) By Sobolev’sinequalityandPoincare’sinequality,weobtain Z Ω z 2n n−2 n−2 n ≤ C Z Ω |∇z|2 + Z Ω z2  ≤ C Z Ω |∇z|2 + C Z Ω z 2 ; 30 where C > 0 independent of r. This,togetherwith(2.5.10) impliesthat Z Ω z2+1 r ≤ C Z Ω |∇z|2 Z Ω z n 2r 2 n + C Z Ω z 2 Z Ω z n 2r 2 n : This isequivalentto Z Ω u2r+1 ≤ C Z Ω |∇ur|2 Z Ω u n 2 2 n + C Z Ω ur 2 Z Ω u n 2 2 n : (2.5.11) This, togetherwithLemma B.0.2 and (2.5.8) impliesthat Z Ω u2r+1 ≤ c1 Z Ω |∇ur|2 + c2; (2.5.12) where c1 := C supt∈(0,Tmax) 􀀀R Ω u n 2  2 n , and c2 := C supt∈(0,Tmax) n􀀀R Ω ur 2 􀀀R Ω u n 2  2 n o . Therefore, there existsapositiveconstant c3 such that d dt Z Ω u2r + Z Ω u2r ≤  2r + 3c1 − 2(2r − 1) r  Z Ω |∇ur|2 + c3: (2.5.13) Wenowhavetochoose  and r > n 4 such that n 4 < r< min ( n 4 4(n − 2) n 2 + n 2χα + 3c1 + 1 ! ; n 2 ) (2.5.14) which ispossibleforany r satisfying n 4 < r< n 4 4(n − 2) n 2 + n 2χα + 3c1 + 1 ! : This, togetherwith(2.5.13), (2.5.14) impliesthatthereexistssome r0 > n 2 such that d dt Z Ω ur0 + Z Ω ur0 ≤ c3: By Gronwall’sinequality,wehave u ∈ L∞ ((0; Tmax); Lr0(Ω)). Wefinallycompletetheproofby applying Theorem 2.5.1. Next weprovethemaintheoremsfortheparabolic-parabolicsystemintwo-andthree-dimensio- nal space. ProofofTheorem 2.1.2 and Theorem 2.1.3. Theorem 2.1.2 and Theorem 2.1.3 are immediatecon- sequences ofLemma 2.4.3, Lemma 2.4.4 and Theorem 2.5.1. 31 CHAPTER 3 BLOW-UPPREVENTIONBYSUB-LOGISTICSOURCESIN2DKELLER-SEGEL SYSTEM The focusofthischapterisonsolutionstoatwo-dimensionalKeller-Segelsystemwithsub-logistic sources. Weshowthatthepresenceofsub-logistictermsisadequatetopreventblow-upphenomena even instronglydegenerateKeller-Segelsystems.Ourproofreliesonseveraltechniques,including parabolic regularitytheoryinOrliczspaces,variationalarguments,interpolationinequalities,and the Moseriterationmethod. 3.1 Introduction Weconsiderthefollowingnonlinearparaboliccross-diffusionpartialdifferentialequations arises fromchemotaxismodels 8>>< >>: ut = ∇ · (D(v)∇u) − ∇· (uS(v)∇v) + f(u) vt = Δv − v + u (KS) in aboundeddomain Ω ⊂ R2 with smoothboundary,where 0 < D ∈ C2([0;∞)) and S ∈ C2([0;∞)) ∩W1,∞((0;∞)) such that S′ ≥ 0; (3.1.1) and f is asmoothfunctiongeneralizingthesub-logisticandsignalproductionsourcerespectively, f(u) = ru −  u2 lnp(u + e) ; with r ∈ R; > 0; and p > 0; (3.1.2) The system(KS) iscomplementedwithnonnegativeinitialconditionsin W1,∞(Ω) not identically zero: u(x; 0) = u0(x); v(x; 0) = v0(x); with x ∈ R; (3.1.3) and homogeneousNeumannboundaryconditionareimposedasfollows: @u @ = @v @ = 0; x ∈ @Ω; t ∈ (0; Tmax); (3.1.4) where  denotes theoutwardnormalvector.Inmoregeneralconditionsfor D and S, asdescribed by (3.1.1) withoutthenon-decreasingrequirementof S, werestudiedin[61]. Itwasproventhat 32 the terms −u2 are sufficientinpreventingblow-upsolutionsbyusingvariationaltechniquesand parabolic regularitytheoryinOrliczspaces.So,itisworthittoinvestigatewhethertheterm ”−uk” forsome k ∈ (1; 2) is sufficienttopreventblow-upsolutions.Thisquestionhasbeen remained opensince2000s,however,previousstudiesfoundthattheterm”−u2” isnotoptimal to preventblow-upsolutions.Indeed,itwasprovedin[71] thatwhen D and S are constantfunc- tions, sub-logisticsources f(u) = ru − μu2 lnp(u+e) where 0 < p< 1 are sufficienttoavoidblow-up solutions. Inthischapter,weapplyparabolicregularityresultsinOrliczspacestoobtainthesim- ilar resultthattheterm −u2 is notoptimalinpreventingblow-upsolutions.Indeed,ourresults indicate thatsolutionstothesystem(KS) undertheconditionsfrom(3.1.1) to(3.1.4) existglobally when theterms −u2 are replacedby −μu2 lnp(u+e) , where 0 < p< 1, withanextraassumptionthat infs≥0 D(s) > 0 or 0 < p< 1/2 without it.Itwasstudiedin[71] thatwhen D and S are constant functions weakerterms −μu2 lnp(u+e) where 0 < p< 1 are sufficienttoavoidblow-upsolutions. The sellingpointofthechapteristheintroductionoftheenergyfunctional y(t) := Z Ω u lnk(u + e) + Z Ω |∇v|2; where thevalueof k is determinedlater.Toestablishanappropriatedifferentialinequalityfor y, we performatediousanalysiscalculation,utilizeinterpolationinequalitiesinSobolevspaces,and employ Moseriterationarguments.Ourapproachisacombinationoftwopreviousideas:thefirst, proposed in[71, Lemma3.2],offersamethodtoobtainauniformboundfor ∥u ln(u)∥ L1(Ω), while the second,describedin[61, Lemma4.5],providesanadditionalargumentforobtainingauniform bound for u ln2(u) L1(Ω). Itisimportanttonotethatusingonlyoneoftheseideasisinsufficient to obtainany u lnk(u) L1(Ω) bounds forsolutions. The chapterisorganizedasfollows.Section 3.2 briefly containsourmainresults.Thelocalwell- posedness ofsolutionsandsomeinterpolationinequalitiesarepresentedinSection 3.3. InSection 3.4, weestablishaprioriestimatesincluding L lnk(L + e), and L2 bounds forsolutions.Finally, the maintheoremsareprovedinSection 3.5. 33 3.2 Maintheorems In thissection,wesummarizetwomaintheoremsfortheexistenceofglobalsolutionstonon- degenerate anddegeneratechemotaxissystems.Letusbeginwiththenondegenatecase: Theorem3.2.1 (Nondegenerate). In additiontotheconditionsfrom (3.1.1) to (3.1.4), weassume that p < 1, and infs≥0 D(s) > 0. Thesystem (KS) possesses aglobalclassicalboundedsolution at alltime. Remark 3.2.1. Our theoremalignswithandstrengthenstheoutcomesof[71, Theorem1.1]by allowing S andDto bearbitraryfunctionssatisfying (3.1.1) rather thanbeingrestrictedtoconstant functions. The degeneratecasearepresentedasfollows: Theorem3.2.2 (Degenerate). If p < 1/2, thenthesystem (KS) with theconditionsfrom (3.1.1) to (3.1.4) admits aglobalclassicalboundedsolutionin Ω × (0;∞). Remark 3.2.2. The theoremrepresentsanadvancementoverthefindingsof[61, Theorem1.4]as it incorporatessub-logisticsourcesinsteadoflogisticones.However,itshouldbenotedthatour resultassumesthenon-decreasingpropertyof S, whereas[61, Theorem1.4]doesnotrequirethis condition. 3.3 Preliminaries The localexistenceanduniquenessofnon-negativeclassicalsolutionstothesystem(KS) can be establishedbyadaptingandadjustingthefixedpointargumentandstandardparabolicregularity theory.Forfurtherdetails,wereferthereaderto[21, 58, 29]. Forconvenience,weadoptLemma 4.1 from[61]. 34 Lemma 3.3.1. Let Ω ⊂ R2 be aboundeddomainwithsmoothboundary,andsuppose r ∈ R and  > 0 and that (3.1.1), (3.1.3), and (3.1.4) hold. Thenthereexist Tmax ∈ (0;∞] and functions 8>>< >>: u ∈ C0 􀀀 ¯Ω × (0; Tmax)  ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  and v ∈ T q>2 C0 ([0; Tmax);W1,q(Ω)) ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  (3.3.1) such that u > 0 and v > 0 in ¯Ω × (0;∞), that (u; v) solves (KS) classically in Ω × (0; Tmax), and that if Tmax < ∞; then lim sup t→Tmax n ∥u∥ L∞(Ω) + ∥u∥ W1,∞(Ω) o = ∞: (3.3.2) 3.4 Aprioriestimates In thissection,weassumethatthesystem(KS) admitsaclassicalsolution (u; v) and amaximal existence time Tmax, subjecttoconditionsgivenby(3.1.1) to(3.1.4), asestablishedinLemma 3.3.1. Toproveourmaintheorems,werelyheavilyonaboundoftheform L lnk(L + e) for solutionsto the systemofequationsin(KS). Theproofutilizesstandardvariationalargumentsandfundamental functional inequalities.Itisworthnotingthatthelogisticdegradationtermsinthefirstequation of (KS), givenby − μu2 lnp(u+e) , effectivelyhandlethecorrespondingcross-diffusioncontribution.To precisely statethisresult,wepresentthefollowinglemma: Lemma 3.4.1. If p ≤ 1 < k< 2 − p, then sup t∈(0,Tmax) Z Ω u lnk (u + e) + |∇v|2 < ∞; and sup t∈(0,Tmax−τ) Z t+τ t Z Ω u2 lnk−p(u + e) +(Δv)2 < ∞; where  = min  1; Tmax 2 . Proof. Wedefine y(t) := Z Ω u lnk (u + e) + 1 2 Z Ω |∇v|2; 35 and differentiate y(·) to obtain y′(t) = Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  ut + Z Ω ∇v · ∇vt := I + J: (3.4.1) Now wemakeuseofthefirstequationof(KS) todealwith I I = Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  (∇ · (D(v)∇u − uS(v)∇v) + f(u)) = −k Z Ω D(v) lnk−1(u + e) u + e |∇u|2 − k(k − 1) Z Ω D(v)u lnk−2(u + e) (u + e)2 |∇u|2 − k Z Ω eD(v) lnk−1(u + e) (u + e)2 |∇u|2 + k Z Ω S(v)u lnk−1(u + e) u + e ∇u · ∇v + k(k − 1) Z Ω S(v)u2 lnk−2(u + e) (u + e)2 ∇u · ∇v + k Z Ω eS(v)u lnk−1(u + e) (u + e)2 ∇u · ∇v + Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  f(u) := X7 i=1 Ii: (3.4.2) Toestimate I4, I5, and I6 from above,weaimtoboundthembyusingtwoterms R Ω u2 lnk−p(u+e), and R Ω(Δv)2. AchievingthisrequiresameticulousapplicationofintegralbypartsandYoung’s inequality.Specifically,wehandle I4 in thefollowingmanner: I4 := k Z Ω eS(v)u lnk−1(u + e) (u + e)2 ∇u · ∇v = k Z Ω S(v)∇1(u) · ∇v; (3.4.3) where 1(l) := Z l 0 s lnk−1(s + e) s + e ≤ l lnk−1(l + e): Weutilizetheintegrationbypartsonequation(3.4.3), takingintoaccountthecondition S′ ≥ 0 36 and applyingYoung’sinequalitytoobtain I4 = −k Z Ω S(v)1(u)Δv − k Z Ω S′(v)1(u)|∇v|2 ≤ c1 Z Ω 1(u)|Δv| ≤  Z Ω (Δv)2 + c2 Z Ω 21 (u) ≤  Z Ω (Δv)2 + c2 Z Ω u2 ln2k−2(u + e) ≤  Z Ω (Δv)2 +  Z Ω u2 lnk−p(u + e) + c3; (3.4.4) where c1 = k ∥S∥ L∞(0,∞),  > 0, c2 > 0 depends on , andthelastinequalitycomesfromthefact that forany  > 0, thereexistapositiveconstant c3 depending on  such that c2u2 ln2k−2(u + e) ≤ u2 lnk−p(u + e) + c3 2k − 2 < k − p: Weapplyasimilarreasoningtohandle I5 and I6. Tobemorespecific,wehave: I5 := k(k − 1) Z Ω S(v)u2 lnk−2(u + e) (u + e)2 ∇u · ∇v = k(k − 1) Z Ω S(v)∇2(u) · ∇v; (3.4.5) where 2(l) := Z l 0 s2 lnk−2(s + e) (s + e)2 ≤ Z l 0 lnk−2(s + e) ≤ 1(l): By usingthesameprocedureto(3.4.4), itfollowsthatforany  > 0, thereexist c4 > 0 depending on  such that I5 ≤  Z Ω (Δv)2 +  Z Ω u2 lnk−p(u + e) + c4: (3.4.6) The term I6 can behandledasfollows I6 := k Z Ω euS(v) lnk−1(u + e) (u + e)2 ∇u · ∇v = −k Z Ω S(v)∇3(u) · ∇v; (3.4.7) 37 where 3(l) := Z l 0 lnk−1(s + e) es (s + e)2 ≤ 1 4 Z l 0 lnk−1(s + e) ≤ l lnk−1(l + e) As theright-handsideof(3.4.7) resemblesthatof(3.4.3), weemploythesamereasoningtodeduce that forany  > 0, thereexist c5 > 0 depending on  such that I6 ≤  Z Ω (Δv)2 +  Z Ω u2 lnk−p(u + e) + c5: (3.4.8) Tohandle I7, wemakeuseofthefactthatforany  > 0, thereexist c() > 0 such that ua1 lnb1(u + e) ≤ ua2 lnb2(u + e) + c(); where a1; a2; b1; b2 are realnumberssuchthat a1 < a2. Thisimpliesthatforany  > 0, thereexist a positiveconstant c7 depending on  such that  lnk (u + e) + k lnk−1 (u + e) u + e  f(u) ≤ ru lnk(u + e) + rk lnk−1(u + e) − u2 lnk−p(u + e) ≤ ( − ) Z Ω u2 lnk−p(u + e) + c7: (3.4.9) Therefore, weobtain: I7 ≤ ( − ) Z Ω u2 lnk−p(u + e) + c7: (3.4.10) Since Ii ≤ 0 for i = 1; 2; 3, andcombinewith(3.4.2), (3.4.4), (3.4.6), (3.4.8) and(3.4.10), forany  > 0, thereexistapositiveconstant c8 depending on  such that I ≤ 3 Z Ω (Δv)2 + (4 − ) Z Ω u2 lnk−p(u + e) + c8: (3.4.11) By integrationbypartsandelementalinequalities,itfollowsthatforany  > 0, thereexist c9 > 0 depending on  such that J := Z Ω ∇v · ∇vt = − Z Ω (Δv)2 − Z Ω |∇v|2 − Z Ω uΔv ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 + 1 2 Z Ω u2 ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 +  Z Ω u2 lnk−p(u + e) + c9: (3.4.12) 38 For any  > 0, thereexistapositiveconstant c10 such that Z Ω u lnk(u + e) ≤  Z Ω u2 lnk−p(u + e) + c10: (3.4.13) By combining(3.4.1), (3.4.11), (3.4.12), and(3.4.13), weobtainthatforany  > 0, thereexist c11 > 0 depending on  such that y′(t) + y(t) + 1 4 Z Ω (Δv)2 +  2 Z Ω u2 lnk−p(u + e) ≤ (3 − 1 4 ) Z Ω (Δv)2 + (6 −  2 ) Z Ω u2 lnk−p(u + e) + c11; (3.4.14) Choose  sufficientlysmall,wehave y′(t) + y(t) ≤ c11: Using Gronwall’sinequalitywiththepreviousequation,itfollowsthat y(t) ≤ max {y(0); c11}. Additionally,wealsohave: 1 4 Z Ω (Δv)2 +  2 Z Ω u2 lnk−p(u + e) ≤ c11 − y′(t): (3.4.15) By integratingthepreviousinequalityfrom t to t +  and usingthefactthat y is bounded,wecan conclude theproof. Remark 3.4.1. The non-decreasingassumptionof S allows ustoobtainauniformboundfor u lnk (u + e) L1(Ω) without usingauniformbound ∥u∥ L1(Ω) as in[61] and[71]. The logisticdegradationterm − μu2 lnp(u+e) can ensuretheboundednessofchemicaldensityfunc- tions, eveninthepresenceofstronglydegeneratediffusionterms.Tostatethisresultprecisely,we present thefollowinglemma. Lemma 3.4.2. If 1 + p 1. Weexaminethenondegeneratediffusionmechanismandobtainboundsfor u and ∇v through a standardtestingprocedure. Lemma 3.4.3. If p < 1, q ≥ 2, S′ ≥ 0, infs≥0 D(s) > 0 and (u; v) is aclassicalsolutionto (KS) in Ω × (0; Tmax) then thereexistsapositiveconstant C such that Z Ω uq(·; t) + Z Ω |∇v(·; t)|2q ≤ C (3.4.16) for all t ∈ (0; Tmax). Proof. Wedefine (t) := 1 q Z Ω uq + 1 2q Z Ω |∇v|2q; and differentiate  to obtain: ′(t) = Z Ω uq−1 [∇ · (D(v)∇u) − ∇· (S(v)u∇v) + f(u)] + Z Ω |∇v|2q−2∇v · ∇(Δv + u − v) := J1 + J2: (3.4.17) By integrationbyparts,wehave J1 = −c1 Z Ω D(v)|∇u q 2 |2 + c2 Z Ω S(v)u q 2∇u q 2 · ∇v + r Z Ω uq −  Z Ω uq+1 lnp(u + e) := J11 + J12 + J13 + J14; (3.4.18) where positiveconstants c1; c2 depends on q. Sinceinf(x,t)∈Ω×(0,T ) D(v(x; t)) > 0, weobtain J11 ≤ −c3 Z Ω |∇u q 2 |2; (3.4.19) for some c3 = c1 inf(x,t)∈Ω×(0,T ) D(v(x; t)). Forany  > 0, thereexistapositiveconstant c4 depending on  such that J12 ≤  Z Ω |∇u q 2 |2 + c4 ∥S∥ L∞(0,∞) Z Ω uq|∇v|2: (3.4.20) 40 Choosing  sufficientlysmallimpliesthat J1 ≤ −c5 Z Ω |∇u q 2 |2 + c6 Z Ω uq|∇v|2 + r Z Ω uq −  Z Ω uq+1 lnp(u + e) ; (3.4.21) where c5 = c3/2 and c6 = c4 ∥S∥ L∞(0,∞). Intreating J2, wemakeuseofthefollowingpointwise identity ∇v · ∇Δv = 1 2 Δ(|∇v|2) − |D2v|2 to obtain J2 = −c7 Z Ω |∇|∇v|q|2 − Z Ω |∇v|2q−2|D2v|2 + Z Ω |∇v|2q−2∇v · ∇u − Z Ω |∇v|2q + c8 Z ∂Ω @|∇v|2 @ |∇v|2q−2; (3.4.22) where c7; c8 are positiveconstantsdependingon q. Theinequality ∂|∇v|2 ∂ν ≤ c|∇v|2 for some c > 0 depending onlyon Ω implies that Z ∂Ω @|∇v|2 @ |∇v|2q−2 dS ≤ c Z ∂Ω |∇v|2q dS: Let g := |∇v|q and applyTraceImbeddingTheorem W1,1(Ω) −→ L1(@Ω) together withYoung’s inequality,weobtainthefollowing c Z ∂Ω g2 dS ≤ C Z Ω g|∇g| + C Z Ω g2 ≤  Z Ω |∇g|2 + c9 Z Ω g2; (3.4.23) for any  > 0 and c9 > 0 depending on . Therefore,wehave Z ∂Ω |∇v|2q dS ≤  Z Ω |∇|∇v|q|2 + c9 Z Ω |∇v|2q: (3.4.24) Applying thepointwiseinequality (Δv)2 ≤ 2|D2v|2 to (3.4.22) andchoosing  = c7/2 yields J2 ≤ −c10 Z Ω |∇|∇v|q|2 − 1 2 Z Ω |∇v|2q−2|Δv|2 + Z Ω |∇v|2q−2∇v · ∇u + c11 Z Ω |∇v|2q = J21 + J22 + J23 + J24; (3.4.25) 41 where c10 = c7/2 and c11 = cc8c9. Byintegrationbypartsandelementalinequalities,weobtain that forany  > 0, thereexistapositiveconstant c12 depending on  such that J23 = Z Ω u|∇v|2q−2∇v · ∇v = − Z Ω |∇v|2q−2Δu − c Z Ω u|∇v|q−1∇|∇v|q · ∇v |∇v| ≤  Z Ω (Δv)2|∇v|2q−2 +  Z Ω |∇|∇v|q|2 + c12 Z Ω u2|∇v|2q−2; (3.4.26) where c is apositiveconstantdependingon q only.Choosing  sufficientlysmall,weobtain J2 ≤ −c13 Z Ω |∇|∇v|q|2 + c11 Z Ω |∇v|2q + c12 Z Ω u2|∇v|2q−2; (3.4.27) where c13 = c10/2. ByYounginequality,forany  > 0, thereexist c14 > 0 depending on  such that: c6 Z Ω uq|∇v|2 + c12 Z Ω u2|∇v|2q−2 ≤  Z Ω |∇v|2q+2 + c14 Z Ω uq+1: (3.4.28) Using theGagliardo-NirenberginequalityinLemma B.0.2 for n = 2 and Lemma 3.4.1, wecan conclude thatthereexistsapositiveconstant cGN such that: Z Ω |∇v|2q+2 ≤ cGN Z Ω |∇|∇v|q|2 Z Ω |∇v|2 + cGN Z Ω |∇v|2 q+1 ≤ c15 Z Ω |∇|∇v|q|2 + c16; (3.4.29) where c15 = cGN supt>0 R Ω |∇v|2 and c16 = cGN 􀀀 supt>0 R Ω |∇v|2 q+1. Thecondition 0 < p< 1 enables ustochoose k ∈ (p; 2 − p), particularlyweselect k = 1 and applyLemma 3.4.1 to obtain the uniformlyboundednessof ∥u ln(u + e)∥ L1(Ω). ThistogetherwithLemma B.0.9 imply thatfor any  > 0, thereexistapositiveconstant c depending on  satisfying Z Ω uq+1 ≤  Z Ω |∇u q 2 |2 Z Ω u ln(u + e) + c Z Ω u q+1 + c ≤ c17 Z Ω |∇u q 2 |2 + c18; (3.4.30) where c17 = supt>0 R Ω u ln(u + e) and c18 > 0 depend on . Combining(3.4.17), (3.4.21), and from (3.4.27) to(3.4.30), andchoosing  sufficientlysmall,weobtain ′(t) ≤ −c19 Z Ω |∇|∇v|q|2 + c20 Z Ω |∇v|2q + r Z Ω uq −  Z Ω uq+1 lnp(u + e) + c21: (3.4.31) 42 For any  > 0, thereexistapositiveconstant c depending on  such that xq ≤ xq+1 lnp(x + e) + c: This impliesthat Z Ω uq ≤  Z Ω uq+1 lnp(u + e) + c22; (3.4.32) where c22 = c|Ω|. ByapplyingLemma B.0.2 and usingthefactthat ∥∇v∥ L2(Ω) is uniformly bounded, andcombiningwithYounginequalityweobtainthatforany  > 0, thereexistapositive constant c23 depending on  and ∥∇v∥ L2(Ω) such that Z Ω |∇|∇v|q|2 ≤ cGN Z Ω |∇|∇v|q|2 q−1 q Z Ω |∇v|2 + cGN Z Ω |∇v|2 q ≤ cGN sup t>0 Z Ω |∇v|2 Z Ω |∇|∇v|q|2 q−1 q + cGN  sup t>0 Z Ω |∇v|2 q ≤  Z Ω |∇|∇v|q|2 + c23: (3.4.33) By combining(3.4.31), (3.4.32), and(3.4.33), andselectinganappropriatevaluefor , wecanfind a postiveconstant c24 depending on  such that ′(t) + (t) ≤ c24. Theproofiscompletedby applying Gronwall’sinequality. When thechemicalconcentrationfunction v is bounded,thedegeneraciesinthediffusionmech- anism areeliminated,thusenablingustoderiveboundsfor u and ∇v. Specifically,wepresentthe following lemma. Lemma 3.4.4. If p < 1/2, q ≥ 2, S′ ≥ 0, and (u; v) is aclassicalsolutionto (KS) in Ω×(0; Tmax) then thereexistsapositiveconstant C such that Z Ω uq(·; t) + Z Ω |∇v(·; t)|2q ≤ C (3.4.34) for all t ∈ (0; Tmax). Proof. Since 0 < p< 1 2 , wecanselectaconstant k ∈ (1 + p; 2 − p). ByutilizingLemma 3.4.1, we obtain sup t∈(0,T−τ) Z t+τ t Z Ω u2 lnk−p(u + e) < ∞: 43 Then, applyingLemma 3.4.2, wededucethat v is globallyboundedintime,implyingthat D(v) ≥ c > 0. UsingthesameargumentasintheproofofLemma 3.4.3, wecanconcludetheproof. It ispossibletoobtainan L∞ bound forsolutionsofequation(KS) byusingLemmaA.1in[53], provided thatwehave Lq0 bounds forsome q0 > 2. However,forthesakeofcompleteness,we present aproofthatusestheMoseriterationmethod[2, 1] toestablishtheiterationprocessfrom Lq0 to L∞. Tothisend,werelyonthefollowinglemma: Lemma 3.4.5. Let (u; v) be aclassicalsolutionof (KS) on (0; Tmax) and Uq := max ( ∥u0∥L∞(Ω); sup t∈(0,Tmax) ∥u(·; t)∥Lq(Ω) ) : If supt∈(0,Tmax) ∥u(·; t)∥Lq(Ω) < ∞for some q >n, thenthereexistsconstants A;B> 0 independent of q such that U2q ≤ (AqB) 1 2qUq: (3.4.35) Proof. The primaryobjectiveistoinitiallyestablishaninequalityoftheform: d dt Z Ω u2q + Z Ω u2q ≤ AqB Z Ω uq 2 ; (3.4.36) where A and B are positiveconstants.WethenproceedtoapplytheMoseriterationtechnique.It is crucialtonotethatthedependenceofalltheconstantson q is trackedcarefully.Multiplyingthe first equationinthesystem(KS) by u2q−1 we obtain 1 2q d dt Z Ω u2q = Z Ω u2q−1ut = Z Ω u2q−1  ∇ · (D(v)∇u) − ∇· (S(v)u∇v) + ru − u2 lnp(u + e)  := I + J + K: (3.4.37) Since thereexist C > 0 such that R Ω uq(·; t) < C for all t ∈ (0; Tmax), Lemma C.1.2 entails that v is globallybounded,whichfurtherimpliesinf(x,t)∈Ω×(0.Tmax) D(v(x; t)) := c1 > 0. Thus,wehave I := −2q − 1 q2 Z Ω D(v)|∇uq|2 ≤ −c1 2q − 1 q2 Z Ω |∇uq|2: (3.4.38) 44 In treating J, J := Z Ω u2q−1∇ · (S(v)u∇v) =  2q − 1 2q Z Ω S(v)∇u2q · ∇v (3.4.39) =  2q − 1 q Z Ω S(v)uq∇uq · ∇v (3.4.40) Lemma (C.1.2) assertsthat v is in L∞ ((0; T);W1,∞(Ω)), whichentailsthat sup 0 0 independent of r;. Substitute  = min n 1 c3q2 ; 1 o into thisyields d dt Z Ω u2q + Z Ω u2q ≤ c5qn+2 Z Ω uq 2 ; (3.4.45) 45 where c5 independent of q. ApplyGronwallinequalityyields Z Ω u2q(·; t) ≤ max  c5qn+2U2q q ; Z Ω u2q 0  This entails ∥u(·; t)∥ L2q(Ω) ≤ max n (c5qn+2) 1 2qUq; |Ω| 1 2q ∥u0∥ L∞(Ω) o ; and furtherimpliesthat U2q ≤ (AqB) 1 2qUq where A = max {c5; |Ω|} and B = n + 2. Theproofof(3.4.35) iscomplete. 3.5 Proofofmaintheorems This sectionfocusesonprovingourmaintheorems,startingwiththenon-degeneratecase. ProofofTheorem 3.2.1. From Lemma 3.4.1 and Lemma 3.4.3, forsomefixed q0 > 2 sup t∈(0,Tmax) Z Ω uq0 + |∇v|2q0 ≤ C < ∞: (3.5.1) By usingLemma C.1.2, wecanconcludethat v belongs to L∞ ((0; Tmax);W1,∞(Ω)). Furthermore, Lemma 3.4.5 implies thatthefollowinginequalityholds U2k+1q0 ≤ 􀀀 A(2kq0)B 1 2k+1q0 U2kq0 (3.5.2) for allintegers k ≥ 0. Aftertakingthelogoftheaboveinequality,wecanuseLemma B.0.10 for the followingsequence. ak = lnA 2k+1q0 + Bk ln 2 2k+1q0 + B ln q0 2k+1q0 One canverifythat X∞ k=0 ak = ln 􀀀 A(2q0)B  q0 : 46 Thus, weobtain U2k+1q0 ≤ A 1 q0 (2q0) B q0 Uq0 (3.5.3) for all k ≥ 1. Send k → ∞ yields U∞ ≤ A 1 q0 (2q0) B q0 Uq0 : (3.5.4) This impliesthat u ∈ L∞ ((0; Tmax); L∞(Ω)). The proofofTheorem 3.2.2 is similartothatofTheorem 3.2.1, withtheadditionalrequirement of showingthatthediffusionmechanismremainsnon-degeneratethroughouttheevolutionofthe system. ProofofTheorem 3.2.2. By usingLemma 3.4.2 and Lemma 3.4.4, itfollowsthatforafixed q0 > 2, we have sup t∈(0,Tmax) Z Ω uq0 + |∇v|2q0 ≤ C < ∞: (3.5.5) Wecannowrepeatthesameargumentsfrom(3.5.2) to(3.5.4) toestablish L∞ bounds for u and v. 47 CHAPTER 4 NONLOCAL; TWOSPECIESWITHTWOCHEMICALS;ANDNONLINEAR BOUNDARYPROBLEMS This chapteraimstoextendthepreviousresearchontheglobalexistenceofsolutionsforchemo- taxis systemsbypresentingfourmainresults.Thefirstresultfocusesontheglobalexistenceof solutions forelliptic-parabolicchemotaxissystemswithlogisticsourcesinthelimitingcase.The second resultexaminestheglobalexistenceofsolutionsintwospecieswithtwochemicalchemo- taxis models.Thethirdresultistoinvestigatetheglobalsolutionsofchemotaxissystemswith nonlocal sources.Finally,weshowthatthequadradicdegradationissufficientlystrongtoprevent blow-up evenfornonlinearNeumannboundarycondition.Theseresultscontributetoourunder- standing oftheglobalexistenceofsolutionsforchemotaxissystemsandhighlightimportantareas of researchwithinthefield. 4.1 Aprioriestimateinthelimitingcases In thissection,weinvestigateontheaprioriestimateforthechemotaxissystemwiththelogistic source f(u) = ru − u2 in Rn where n ≥ 3. 8 >>< >>: ut = ∇ · (D(v)∇u) − ∇· (uS(v)∇u) + f(u) vt = Δv + u − v (4.1.1) Wealsohavetheglobalboundednesspropertyforthesolutiontotheparabolic-ellipticsystem when  = n−2 n  , n ≥ 3 and f(u) = au−u2. Hereweprovideashorterproofbutsimilartothe result in[23]. Theorem4.1.1. If  = n−2 n  and (u; v) is aclassicalsolutionof (4.1.1) on Ω × (0; Tmax) with maximal existencetime Tmax ∈ (0;∞], then sup t∈(0,Tmax)  ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω)  < ∞: 48 Proof. Multiplying thefirstequationinthesystem(4.1.1) by u2r−1 yields 1 2r d dt Z Ω u2r = Z Ω u2r−1ut = Z Ω u2r−1 [Δu − ∇(u∇v) + f(u)] = −2r − 1 r2 Z Ω |∇ur|2 −  2r − 1 2r Z Ω u2rΔv + Z Ω u2r−1f(u) = −2r − 1 r2 Z Ω |∇ur|2 + 2r − 1 2r Z Ω u2r( u −  v) + a Z Ω u2r −  Z Ω u2r+1: (4.1.2) Since v ≥ 0, wehave d dt Z Ω u2r ≤ −2(2r − 1) r Z Ω |∇ur|2 − [2r −  (2r − 1)] Z Ω u2r+1 + 2ra Z Ω u2r: (4.1.3) Plug  = n−2 n  into thelasttermof(2.4.3), wehave −[2r −  (2r − 1)] Z Ω u2r+1 =   4 n r − 1 Z Ω u2r+1 Now,wechoose r = n 4 to obtain d dt Z Ω u n 2 + Z Ω u n 2 ≤ −4(n − 2) n Z Ω |∇u n 4 |2 + na + 2 2 Z Ω u n 2 ; By applyingGNinequality,thenYounginequalityandfinallymakinguseofsupt∈(0,T ) R Ω u < ∞, we obtainthatforeveryarbitrarysmall  > 0, thereexists c = c() > 0 such that Z Ω u n 2 ≤  Z Ω |∇u n 4 |2 + c: Therefore, wechoose  sufficientlysmalltoobtain d dt Z Ω u n 2 + Z Ω u n 2 ≤ c: By Gronwallinequality,wehave sup t∈(0,Tmax) Z Ω u n 2 (x; t) dx ≤ c 49 For every  > 0, thereexists r such that  􀀀 4 nr − 1  < , wehave d dt Z Ω u2r ≤ −2(2r − 1) r Z Ω |∇ur|2 +  Z Ω u2r+1: (4.1.4) WeapplyGN-inequalitytoobtain Z Ω u2r+1 ≤ C Z Ω |∇ur|2 Z Ω u n 2 2 n + Z Ω u 2r+1 ≤ C Z Ω |∇ur|2 + C; (4.1.5) where C is independentof r. Thusfrom(4.1.4) and(4.1.5), wehave d dt Z Ω u2r ≤  C − 2(2r − 1) r Z Ω |∇ur|2 Now,wechoose  such that C − 2(2r − 1) r ≤ 0 which ispossiblesincetheinequality   4 n r − 1  < < 2(2r − 1) Cr holds when n 4 < r< n 4  4(n − 2) Cn + 1  : Therefore, thereexists p > n 2 such that sup t∈(0,Tmax) Z Ω up < ∞ which furtherimplies u ∈ L∞ ((0;∞); L∞(Ω)). 4.2 Nonlocalproblems In thissection,westudysomechemotaxismodelsinvolvingnonlocaltermsasthefollowing 8>>< >>: ut = Δu − ∇· (u∇v) + f(u) vt = Δv − v + u (4.2.1) where f(u) = ru − u 􀀀R Ω up q with positiveparameters r;;p;q . 50 Theorem4.2.1. The problem (4.2.1) with parameters  = 0, p > n 2 and q > 1 + 4 2p−n possesses a global classicalsolution (u; v). Proof. Multiplying thefirstequationinthesystem(4.1.1) by up−1 yields 1 p d dt Z Ω up = Z Ω up−1ut = Z Ω up−1 [Δu − ∇(u∇v) + f(u)] = −4(p − 1) p2 Z Ω |∇u p 2 |2 −  p − 1 p Z Ω upΔv + Z Ω up−1f(u) = −4(p − 1) p2 Z Ω |∇u p 2 |2 + p − 1 p Z Ω up(u − v) + r Z Ω up −  Z Ω up q+1 : (4.2.2) Since v ≥ 0, wehave d dt Z Ω up ≤ −p − 1 p Z Ω |∇u p 2 |2 + (p − 1) Z Ω up+1 + pr Z Ω up −  Z Ω up q+1 : (4.2.3) Now wemakeuseofGigliardo-Neirenberginequalitytoobtain Z Ω up+1 ≤ CGN Z Ω |∇u p 2 |2  n 2p Z Ω up 2p−n+2 2 + CGN Z Ω u p+1 : (4.2.4) Since n 2p < 1, weapplyYoung’sinequalitytothefirsttermof(4.2.4), toobtain (p − 1) Z Ω up+1 ≤  Z Ω |∇up/2|2 + c Z Ω up 2(2p−n+4) 2p−n + CGNMp+1; (4.2.5) where  > 0 will bedeterminedlater, c = c() > 0, and M := sup t∈(0,T ) Z Ω u(x; t) dx < ∞: From (4.2.3) and(4.2.5), weimply d dt Z Ω up ≤   − p − 1 p Z Ω |∇u p 2 |2 + pr Z Ω up + c Z Ω up 2(2p−n+4) 2p−n −  Z Ω up q+1 + CGNMp+1: (4.2.6) 51 Now wechoose  < p−1 p and denote y(t) := Z Ω up; and g(s) := prs + s 2(2p−n+4) 2p−n − sq+1 + CGNMp+1. Wehavethefollowingdifferentialinequality y′(t) ≤ g(y(t)): The condition q + 1 > 2(2p − n + 4) 2p − n is equivalentto q > 1 + 4 2p − n : Thus, theequation g(s) =0 has apositivesolution s0 such thatforall s >s0 we have g(s) < 0. Finally,wefindthat y′(t) < 0 when y(t) > s0, andthereforemaximumprincipleimpliesthat y(t) is boundedglobally. 4.3 Two-specieschemotaxissystemwithtwochemicalswithsub-logisticsourcesin2d This sectionaimstostudytheglobalexistenceandboundednessofsolutionsinatwo-species chemotaxis systemwithtwochemicalsandsub-logisticsources.Theappearanceofasub-logistic source inonlyonecelldensityequationeffectivelypreventstheoccurrenceofblow-upsolutions, even infullyparabolicchemotaxissystems. 4.3.1 Introduction Weconsideramodelinvolvingtheinteractionoftwospeciesthroughchemotaxis,whereeach species emitsasignalthatinfluencesthemovementoftheotherspecies.Specifically,westudythe following PDEinaopenboundeddomain Ω ⊂ R2 8>>>>>>>>>>< >>>>>>>>>>: ut = Δu − ∇· (u∇v) + f(u) vt = Δv − v + w; wt = Δw − ∇· (w∇z) zt = Δz − z + u (4.3.1) 52 where  ∈ {0; 1}, and f(u) = ru− μu2 lnp(u+e) , with r ∈ R,  ≥ 0, and p ≥ 0, underthehomogeneous Neumann boundarycondition @u @ = @v @ = 0; (x; t) ∈ @Ω × (0; Tmax); (4.3.2) where Tmax ∈ (0;∞] is themaximalexistencetimeforclassicalsolutions. The effectoflogisticsourcesorsub-logisticsourcesonblow-uppreventionintwo-species chemotaxis modelsisquitelimited,especiallywhenthelogisticsourcesappearonlyinthefirst equation ofthesystem(4.3.1). In[60], theauthorsstudytheglobalexistenceandlongtimebehavior of solutionstothefollowingsystem 8>>>>>>>>>>< >>>>>>>>>>: ut = Δu − ∇· (u∇v) + r1u − 1u2 vt = Δv − v + w; wt = Δw − ∇· (w∇z) + r2w − 2w2 0 =Δz − z + u; (4.3.3) with r1; r2 ∈ R and 1; 2 > 0: It wasshownthatif 12 is sufficientlylargethenallsolutionsto (4.3.3) areglobalandboundedforany n ≥ 1. Forfurtherstudiesonglobalexistenceandequilib- rium solutionsfortwospecieswithlogisticsourcesappearingintwocelldensityequations,readers can referto[3, 55, 11]. Infact,theanalysisframeworktoprovetheglobalexistenceofsolutions to (4.3.3) forsufficientlylarge 12 is similartotheoneforonespecies[64]. However,therehas no resultsofarconsideringthepresenceofalogisticsourceinonlyonespecies.Ourpurposeisto address thattheappearanceofthesub-logisticsourcesinonespeciescaneffectivelyeliminatethe occurrence offinitetimeorinfinitetimeblow-upsolutionsintwodimensionaldomains.Precisely, we havethefollowingtheorem: Theorem4.3.1. Assume that  ∈ {0; 1}, f(u) = ru − μu2 lnp(u+e) , where r;> 0, and p ∈ [0; 1) and nonnegative initialdata u0;w0 ∈ C0,α(¯Ω) for some ∈ (0; 1) when  = 0 and u0;w0 ∈ C0(¯Ω) and v0; z0 ∈ W1,∞(Ω) when  = 1. Thenthereexistsauniquequadruple (u; v;w;z) of nonnegative 53 functions u ∈ C0 􀀀 ¯Ω × [0;∞)  ∩ C2,1 􀀀 ¯Ω × (0;∞)  ; v ∈ C2,1 􀀀 ¯Ω × (0;∞)  ; w ∈ C0 􀀀 ¯Ω × [0;∞)  ∩ C2,1 􀀀 ¯Ω × (0;∞)  and z ∈ C2,1 􀀀 ¯Ω × (0;∞)  ; which solvethesystem (4.3.1) in theclassicalpointwisesensein Ω × (0;∞). Moreover, sup t∈(0,∞) n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) + ∥w(·; t)∥ L∞(Ω) + ∥z(·; t)∥ W1,∞(Ω) o < ∞: (4.3.4) In thefollowingsections,wewillbrieflyrecallthelocalwell-posednessresultsforsolutions to thesystem(4.3.1) inSection 4.3.2, andexplorethemechanismsbehindblow-uppreventionby sub-logistic sourcesinSection 4.3.3. 4.3.2 Localexistence The localexistenceofsolutionstothesystem(4.3.1) underhomogeneousNeumannboundary conditions canbeprovedbyadaptingapproachesthatarewell-establishedinthecontextchemotaxis models withlogisticsources.Firstly,weestablishthelocalexistenceofsolutionsforparabolic- elliptic chemotaxismodelsbyadapting[59][Theorem 2.1]. Lemma 4.3.2. Suppose that  = 0, ∈ (0; 1) and u0 and w0 arenonnegativefunctionsin C0,α(¯Ω). Then thereexist Tmax ∈ (0;∞] and auniquequadruple (u; v;w;z) of nonnegativefunctionsfrom C0(¯Ω ×(0; Tmax))∩C2,1(¯Ω ×(0; Tmax)) solving (4.3.1) under boundarycondition (4.3.2) classically in Ω × (0; Tmax). Moreover,if Tmax < ∞, then lim sup t→Tmax  ∥u(·; t)∥ L∞(Ω) + ∥w(·; t)∥ L∞(Ω)  = ∞: (4.3.5) Secondly,onecanadaptandmodifytheproofof[64][Lemma 1.1]toobtainthelocalexistence of solutionsforfullyparabolicmodels. Lemma 4.3.3. Suppose that  = 1, and (u0; v0;w0; z0) ∈ C0(¯Ω)×W1,∞(Ω)×C0(¯Ω)×W1,∞(Ω) such that u0; v0;w0; z0 arenonnegative.Thenthereexist Tmax ∈ (0;∞] and auniquequadruple 54 (u; v;w;z) of nonnegativefunctions u;w ∈ C0(¯Ω × (0; Tmax)) ∩ C2,1(¯Ω × (0; Tmax)) (4.3.6) v;z ∈ C0(¯Ω × (0; Tmax)) ∩ C2,1(¯Ω × (0; Tmax)) × L∞ loc 􀀀 [0; Tmax);W1,∞(Ω)  solving (4.3.1) under boundarycondition (4.3.2) classically in Ω × (0; Tmax). Moreover,if Tmax < ∞, then lim sup t→Tmax  ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) + ∥w(·; t)∥ L∞(Ω) + ∥z(·; t)∥ W1,∞(Ω)  = ∞: (4.3.7) 4.3.3 Globalboundednesswithsub-logisticsources.ProofofTheorem 4.3.1 The subsequentlemmaholdsacentralpositioninthissection,servingasacornerstoneofour work. Anoteworthyinnovationintroducedhereinisthefunctionaldescribedby(4.3.9), withthe specific valuesofthepositiveparameters A and B to bedeterminedsubsequentlyintheanalysis. It isnotablethat,withinthecontextofinequality(4.3.24), weidentifyauniquechoicefor A that depends ontheparameter , resultinginthenonpositivityofthefirsttermontheright-handsideof (4.3.24). Lemma 4.3.4. Under theassumptionsasinTheorem 4.3.1, thereexistsapositiveconstant C such that Z Ω u(·; t) ln(u(·; t)+e)+ Z Ω w(·; t) ln(w(·; t)+e)+ Z Ω |∇v(·; t)|2+ Z Ω |∇z(·; t)|2 < C; (4.3.8) for all t ∈ (0; Tmax). Remark 4.3.1. Lemma 4.3.4 continues toholdinsmoothboundeddomainswitharbitrarydimen- sion. Proof. Wedefine y(t) := Z Ω u ln(u + e) + Z Ω w ln(w + e) + A 2 Z Ω |∇v|2 + B 2 Z Ω |∇z|2; (4.3.9) 55 where A := 2, B :=  + 1 4ϵ and  := min n μ 4 ; 1 3GGN 􀀀R Ω w0 + e|Ω| −1 o . Differentiating y in time, we obtain y′(t) + y(t) = Z Ω  ln(u + e) + u u + e  Δu − ∇· (u∇v) + ru − u2 lnp(u + e)  Z Ω  ln(w + e) + w w + e  (Δw − ∇· (w∇z)) + A Z Ω ∇v · ∇(Δv − v + w) + B Z Ω ∇z · ∇(Δz − z + u) := I1 + I2 + I3 + I4: (4.3.10) In case  = 1, weuseintegrationbypartstoobtain: I1 = − Z Ω |∇u|2 u + e − Z Ω e|∇u|2 (u + e)2 + Z Ω  u u + e + eu (u + e)2  ∇u · ∇v + Z Ω  ln(u + e) + u u + e  ru − u2 lnp(u + e)  : (4.3.11) Let usdefine (u) := Z u 0  s s + e + es (s + e)2  ds; we seethat (u) ≤ u. When  = 0, byintegrationbypartsandelementaryinequality,wemake use ofthesecondequationstoobtainthat: Z Ω  u u + e + eu (u + e)2  ∇u · ∇v = Z Ω ∇(u) · ∇v = − Z Ω (u)Δv = Z Ω (u)(w − v) ≤ Z Ω uw ≤  Z Ω w2 + 1 4 Z Ω u2 ≤  Z Ω w2 +  Z Ω u2 ln1−p(u + e) + c; (4.3.12) where thelastinequalitycomesfromthefactthatforany  > 0, thereexists C > 0 depending on  such that u2 ≤ u2 ln1−p(u + e) + C(); 0 ≤ p < 1: 56 Weapplysimilarargumentincase  = 1 to obtain Z Ω  u u + e + eu (u + e)2  ∇u · ∇v = − Z Ω (u)Δv ≤  Z Ω (Δv)2 + 1 4 Z Ω 2(u) ≤  Z Ω (Δv)2 + 1 4 Z Ω u2 ≤  Z Ω (Δv)2 +  Z Ω u2 ln1−p(u + e) + c; (4.3.13) One canverifythatthereexists c > 0 depending on  such that Z Ω  ln(u + e) + u u + e  ru − u2 lnp(u + e)  ≤ ( − ) Z Ω u2 ln1−p(u + e) + c: (4.3.14) From (4.3.11) to(4.3.14), weobtainthat I1 ≤ (2 − ) Z Ω u2 ln1−p(u + e) +  Z Ω w2 + c; for  = 0; (4.3.15) and, I1 ≤ (2 − ) Z Ω u2 ln1−p(u + e) +  Z Ω (Δv)2 + c; for  = 1: (4.3.16) By similararguments,onecanalsoobtainthat I2 ≤ − Z Ω |∇w|2 w + e +  Z Ω w2 +  Z Ω u2 ln1−p(u + e) + c; for  = 0; (4.3.17) and I2 ≤ − Z Ω |∇w|2 w + e +  Z Ω w2 + 1 4 Z Ω (Δz)2; for  = 1: (4.3.18) By integrationbypartsandelementaryinequalities,wehave I3 = −A Z Ω (Δv)2 − A Z Ω |∇v|2 − A Z Ω wΔv ≤  ( A2 4 − A) Z Ω (Δv)2 − A Z Ω |∇v|2 +  Z Ω w2; (4.3.19) 57 and I4 = −B Z Ω (Δz)2 − B Z Ω |∇z|2 − B Z Ω uΔz ≤  ( − B) Z Ω (Δz)2 − B Z Ω |∇z|2 +  B2 4 Z Ω u2 ≤  ( − B) Z Ω (Δz)2 − B Z Ω |∇z|2 +   Z Ω u2 ln1−p(u + e) + c: (4.3.20) One canverifythat Z Ω u ln(u + e) + Z Ω w ln(w + e) ≤  Z Ω u2 ln1−p(u + e) +  Z Ω w2 + c: (4.3.21) From (4.3.15) to(4.3.21), wehave y′(t) + y(t) ≤ − Z Ω |∇w|2 w + e + (4 − ) Z Ω u2 ln1−p(u + e) +3 Z Ω w2 +   A2 4 +  − A Z Ω (Δv)2 +    + 1 4 − B Z Ω (Δz)2 + c ≤ − Z Ω |∇w|2 w + e + (4 − ) Z Ω u2 ln1−p(u + e) +3 Z Ω w2; (4.3.22) where thelastinequalitycomesfromthefactthat A2 4ϵ +  − A = 0 and B =  + 1 4ϵ . Thethirdterm can becontrolledbyGagliardo–Nirenberginterpolationinequality 3 Z Ω w2 ≤ 3CGN Z Ω |∇w|2 w + e Z Ω (w + e) +3CGN Z Ω (w + e) 2 ≤ 3CGN Z Ω w0 + e|Ω| Z Ω |∇w|2 w + e + 3CGN Z Ω w0 + e|Ω| 2 : (4.3.23) From (4.3.22) and(4.3.23), wehave y′(t) + y(t) ≤  3CGN Z Ω w0 + e|Ω|  − 1  Z Ω |∇w|2 w + e + (4 − ) Z Ω u2 ln1−p(u + e) + c: (4.3.24) Given theinequalities 3CGN 􀀀R Ω w0 + e|Ω|  −1 ≤ 0 and 4− ≤ 0, wecandeducefrom(4.3.24) that y′(t) + y(t) ≤ c. FinallywemakeuseofGronwall’sinequalitytocompletetheproof. Wearenowreadytoprovethemaintheorem. 58 ProofofTheorem 4.3.1. Weemploytheargumentsfrom[54][Lemma 4.2]withsomemodifica- tions, andleverageLemma 4.3.4 to derivethefollowinginequalityforall t ∈ (0; Tmax) ∥u(·; t)∥ L2(Ω) + ∥w(·; t)∥ L2(Ω) < C: Subsequently,weapplyMoser-typeiterations,akinto[55][Lemma 3.2]toestablishthebounded- ness of u and w in Ω×(0; Tmax). Combiningthiswith(4.3.5) when  = 0 and (4.3.7) when  = 1, we concludethat Tmax = ∞. Employingellipticregularityfor  = 0, andparabolicregularityfor  = 1, wehavethatsupt>0  ∥v(·; t)∥ W1,∞(Ω) + ∥z(·; t)∥ W1,∞(Ω)  < ∞. Consequently,wederive (4.3.4), therebycompletingtheproof. 4.4 Blow-uppreventionbysub-logisticsourcesundervanishingNeumannboundarycondi- tion This sectioninvestigatestheglobalexistenceofsolutionstoKeller-Segelsystemswithsub- logistic sourcesusingthetestfunctionmethod.Priorworkby[71] demonstratedthatsub-logistic sources f(u) = ru −  u2 lnp(u+e) with p ∈ (0; 1) can preventblow-upsolutionsforthe2Dmini- mal Keller-Segelchemotaxismodel.Ourstudyextendsthisresultbyshowingthatwhen p = 1, sub-logistic sourcescanstillpreventtheoccurrenceoffinitetimeblow-upsolutions.Addition- ally,weprovideaconciseproofforaresultpreviouslyprovenin[9] thattheequi-integrabilityof R Ω u n 2 (·; t) t∈(0,Tmax) can avoidblow-up. 4.4.1 Introduction In thissection,weconsiderthefollowingchemotaxismodelwithsub-logisticsourcesina bounded domainwithsmoothboundary Ω ⊂ Rn, where n ≥ 2: 8>>< >>: ut = Δu − ∇· (u∇v) + f(u) 0 =Δv + u − v; (4.4.1) where f is asmoothfunctiongeneralizingthesub-logisticsource, f(u) = ru −  u2 lnp(u + e) ; with r ∈ R; > 0; and p > 0: (4.4.2) 59 The system(4.4.1) iscomplementedwithnonnegativeinitialconditionsinW1,∞(Ω) not identically zero: u(x; 0) = u0(x); v(x; 0) = v0(x); with x ∈ Ω; (4.4.3) and homogeneousNeumannboundaryconditionareimposedasfollows: @u @ = @v @ = 0; x ∈ @Ω; t ∈ (0; Tmax); (4.4.4) where  denotes theoutwardnormalvector. The logisticsources, f(u) := ru − u2, wasintroducedandstudiedin[58] thatif  > n−2 n then solutionsexistgloballyandareboundedatalltimeinaconvexopenboundeddomain Ω ⊂ Rn where n ≥ 2. Inorderword,if  is sufficientlylarge,thenthequadraticterm −u2 ensures no occurrence ofblow-upsolutionsintwospacialdimensionaldomain.Thisleadstoanaturalques- tion thatwhethertheterm”−u2” isoptimaltopreventblow-upsolutions.However,ithasbeen discovered in[71] thattheanswerisnegative.Tobespecific,the”weaker”term − μu2 lnp(u+e) for 0 < p< 1 is sufficienttoavoidblow-upsolutionsforbothelliptic-parabolicandfullyparabolic minimal Keller-Segelchemotaxismodelsinatwospacialdimensionaldomain.Ourmainwork improve thepreviousfindingbyshowingthat p = 1 can preventblow-upsolutionsofthesystem (4.4.1). Our analysisreliesonatestfunctionmethodandMoseriterationtechnique.Itisprovedin [9] thatifthefamilyof R Ω u n 2 (·; t) t∈(0,Tmax) is equi-integrable,thensolutionsof(4.4.1) when f ≡ 0 exist globallyandremainboundedatalltime.Inthissection,wegiveanothershorter proof inProposition 4.4.1 for thatresultaswellasindicatethattheequi-integrabilityisnotoptimal to preventblow-upthanktodelaVallée-PoussinTheorem.Thereafter,wetrytofindasuitable functional andestablishadifferentialinequalitytoobtainaprioriestimateforsolutionsof(4.4.1) thank tothepresenceofthesub-logisticquadraticdegradationterm”− u2 ln(u+e) ”. Indeed,thekey 60 milestone inthisstudyisthechoiceofthefollowingfunctional: y(t) = Z Ω u(·; t) ln(ln(u(·; t) + e)); (4.4.5) which enablesustocontroltheintegral R Ω u2 ln(u+e) to establishaappropriatedifferentialinequality. One canalsotrytoexamineafunctional yk(t) = Z Ω u(·; t) lnk(u(·; t) + e) to findanappropriate k, however,thereisnosuitable k satisfying theconditionsthat  can be arbitrary small.Inorderword,thismethodleadstothechoiceof k, butitdoesrequirethelargeness assumption for . Sothefunctional(4.4.5) enablesustoovercomethatobstacletoproveourmain theorem asfollows: Theorem4.4.1. Let  > 0, and Ω ⊂ R2 be aboundeddomainwithsmoothboundary.Thesystem (4.4.1) under theassumptions (4.4.2), (4.4.3), and (4.4.4) admits aglobalboundedsolutionin Ω × (0;∞) . Remark 4.4.1. Theorem 4.4.1 is aspecialcaseof[72][Remark 1.1(ii)] . 4.4.2 Preliminaries The localexistenceanduniquenessofnon-negativeclassicalsolutionstothesystem(4.4.1) can be establishedbyadaptingandadjustingthefixedpointargumentandstandardparabolicregularity theory.Forfurtherdetails,wereferthereaderto[21, 29, 58]. Forconvenience,weadoptLemma 4.1 from[61]. Lemma 4.4.2. Let Ω ⊂ Rn, where n ≥ 2 be aboundeddomainwithsmoothboundary,andsuppose r ∈ R,  > 0, theconditions (4.4.3), and (2.1.2) hold. Thenthereexist Tmax ∈ (0;∞] and functions 8>>< >>: u ∈ C0 􀀀 ¯Ω × [0; Tmax)  ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  and v ∈ T q>2 C0 ([0; Tmax);W1,q(Ω)) ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  (4.4.6) 61 such that u > 0 and v > 0 in ¯Ω ×(0;∞), that (u; v) solves (4.4.1) classically in Ω×(0; Tmax), and that if Tmax < ∞, then lim sup t→Tmax n ∥u∥ L∞(Ω) + ∥u∥ W1,∞(Ω) o = ∞: (4.4.7) 4.4.3 Aprioriestimatesandproofofmaintheorem In thissection, (u; v) is aclassicalsolutionsasdefinedinLemma 4.4.2 to thesystem(4.4.1) with p = 1. Ouraimistoestablishaprioriestimateforthesolutions.Whilethemethodin[47] and [71] reliesonthe L1-estimate of u and theabsorptionof − R Ω |∇u 1 2 |2 to obtaina L ln L uniform bound, wetakeadvantageoftheterm − u2 ln(u+e) to obtainaweaker L ln ln L uniform bound.This result isaspecialcaseof[72][Remark 1.1(ii)].Noticethatwehaveadoptedandmodifiedthe argumentof[72] fortheglobalexistenceofsolutionsincase p = 1. Lemma 4.4.3. Thereexists C = C(u0; v0; |Ω|; ) > 0 such that sup t∈(0,Tmax) Z Ω u(·; t) ln(ln(u(·; t) + e)) ≤ C: (4.4.8) Proof. Wedefine y(t) = R Ω u ln(ln(u + e)) and differentiate y to obtain y′(t) = Z Ω  ln(ln(u + e)) + u (u + e) ln(u + e)  ut = Z Ω  ln(ln(u + e)) + u (u + e) ln(u + e)  Δu − ∇· (u∇v) + ru −  u2 ln(u + e)  = − Z Ω ∇  ln(ln(u + e)) + u (u + e) ln(u + e)  · ∇u + Z Ω u∇  ln(ln(u + e)) + u (u + e) ln(u + e)  · ∇v + Z Ω  ln(ln(u + e)) + u (u + e) ln(u + e)  ru −  u2 ln(u + e)  := I + J + K (4.4.9) 62 By integrationbyparts,wehave I = − Z Ω ∇  ln(ln(u + e)) + u (u + e) ln(u + e)  · ∇u = − Z Ω  1 (u + e) ln(u + e) + e ln(u + e) − u (u + e)2 ln2(u + e)  |∇u|2 = − Z Ω u ln(u + e) +2e ln(u + e) − u (u + e)2 ln2(u + e) |∇u|2 ≤ 0: (4.4.10) Similarly,wehave J = Z Ω u∇  ln(ln(u + e)) + u (u + e) ln(u + e)  · ∇v = Z Ω u2(ln(u + e) − 1) +2eu ln(u + e) (u + e)2 ln2(u + e) ∇u · ∇v = Z Ω ∇(u) · ∇v = Z Ω (u)(u − v) ≤ Z Ω u(u); (4.4.11) where 0 ≤ (u) := Z u 0 s2(ln(s + e) − 1) +2es ln(s + e) (s + e)2 ln2(s + e) ds ≤ Z u 0 1 ln(s + e) ds: (4.4.12) Thus, weobtain J ≤ Z Ω u Z u 0 1 ln(s + e) ds: (4.4.13) By L’Hospitallemma,wehave lim u→∞ R u 0 1 ln(s+e) ds u ln(ln(u+e)) ln(u+e) = lim u→∞ ln(u + e) ln(u + e) ln(ln(u + e)) + u u+e − u u+e ln(ln(u + e)) = 0: (4.4.14) Therefore, forany  > 0, thereexist N depending on  such thatfor u >N, wehave Z u 0 1 ln(s + e) ds ≤ u ln(ln(u + e)) ln(u + e) : (4.4.15) This leadsto Z Ω u Z u 0 1 ln(s + e) ds = Z u≤N u Z u 0 1 ln(s + e) ds + Z u>N u Z u 0 1 ln(s + e) ds ≤  Z Ω u2 ln(ln(u + e)) ln(u + e) + c (4.4.16) 63 where c = N2|Ω|. From(4.4.13) and(4.4.16), wehave J ≤  Z Ω u2 ln(ln(u + e)) ln(u + e) + c: (4.4.17) One canverifythatforany  > 0, thereexist C() > 0 such that K = Z Ω  ln(ln(u + e)) + u (u + e) ln(u + e)  ru −  u2 ln(u + e)  ≤ ( − ) Z Ω u2 ln(ln(u + e)) ln(u + e) + c (4.4.18) and y(t) ≤  Z Ω u2 ln(ln(u + e)) ln(u + e) + c: (4.4.19) Collect (4.4.9), (4.4.10), (4.4.13), (4.4.17),(4.4.18), and(4.4.19), wehave y′(t) + y(t) ≤ (3 − ) Z Ω u2 ln(ln(u + e)) ln(u + e) + c: (4.4.20) Wechoose  sufficientlysmallandapplyGronwall’sinequalitytoimply y(t) ≤ C for all t > 0. Let usrecalldelaVallée-PoussinTheorem Theorem4.4.4 (de laVallée-Poussin). The family {Xα} α∈A ⊂ L1() is equi-integrableifandonly if thereexistsanon-negativeincreasingconvexfunction G(t) such that lim t→∞ G(t) t = ∞ and sup α Z Ω G(Xα) < ∞: Thank toTheorem 4.4.4, theequi-integrabilityof R Ω u n 2 (·; t) < ∞ t∈(0,Tmax) is equivalentto supt∈(0,Tmax) R Ω G(u n 2 (·; t)) < ∞ for somenon-negativeincreasingconvexfunctionsuchthat lim s→∞ G(s) s = ∞: However,theconvexityconditionisnotnecessary,whichmeansthattheequi-integrablecondition can berelaxed.Indeed,followingpropositiongivesusthe Lq bounds, where q > n 2 for solutions without theconvexityassumption. 64 Proposition4.4.1. Let Ω ⊂ Rn, where n ≥ 2, beaboundeddomainwithsmoothboundary,and f ∈ C2([0;∞)) such that f(s) ≤ c(s2 + 1) for all s ≥ 0, where c > 0. Assumethat (u; v) is a classicalsolutionasinLemma 4.4.2 of (4.4.1) on Ω × (0; Tmax) with maximalexistencetime Tmax ∈ (0;∞]. Ifthereexistsanonnegativeincreasingfunction G such that lim t→∞ G(s) s = ∞ and sup t∈(0,Tmax) Z Ω G(u n 2 (·; t)) < ∞; then forany q > n 2 we have sup t∈(0,Tmax) Z Ω uq(·; t) < ∞: Proof. Wedefine (t) := 1 q Z Ω uq; where q > n 2 , anddifferentiate  to obtain ′(t) = Z Ω uq−1[Δu − ∇· (u∇v) + f(u)] = −c1 Z Ω |∇u q 2 |2 + c2 Z Ω u q 2∇u q 2 · ∇v + c Z Ω uq+1 + uq−1 = I + J + K; (4.4.21) where c1; c2 are positivedependingonlyon q. Wemakeuseofintegrationbypartsandthesecond equation of(4.4.1) toobtain J := c2 Z Ω u q 2∇u q 2 · ∇v = −c3 Z Ω uqΔv = −c3 Z Ω uq(v − u) ≤ c3 Z Ω uq+1; (4.4.22) where c3 is positivedependingonlyon q. From(4.4.21), (4.4.22), togetherwithYounginequality, imply thatthereexists c4 = c4(q) > 0, and c5 = c5(q; |Ω|) > 0 such that ′(t) + (t) ≤ −c1 Z Ω |∇u q 2 |2 + c4 Z Ω uq+1 + c5: (4.4.23) WemakeuseofLemma B.0.9 to obtainthatthereexist C > 0 such thatforany  > 0, thereexists c6 = c6() > 0 such that c5 Z Ω uq+1 ≤  Z Ω |∇u q 2 |2 Z Ω G(u n 2 ) 2 n + C Z Ω u q+1 + c6 Z Ω u 65 This, togetherwiththeuniformboundedconditionof R Ω G(u n 2 (·; t)) imply that c4 Z Ω uq+1 ≤ c7 Z Ω |∇u q 2 |2 + c8; (4.4.24) where c7 is positiveindependentof  and c8 = c8() > 0. From(4.4.21) to(4.4.24), weobtainthat ′(t) + (t) ≤ (c7 − c1) Z Ω |∇u q 2 |2 + c9; (4.4.25) where c9 = c5 + c8. Theproofisnowcompletedbychoosing  < c1 c7 and applyingGronwall’s inequality. Wearenowreadytoprovethemainresult. ProofofTheorem 4.4.1. From Lemma 4.4.3, weobtainthatthereexists C1 > 0 such that sup t∈(0,Tmax) Z Ω G(u(·; t)) ≤ C1; where G(s) := s ln(ln(s + e)), satisfyingallconditionsofProposition 4.4.1. Therefore,wecan apply Proposition 4.4.1 to deducethatforany q > 1 there exists C2 = C2(q) > 0 such that sup t∈(0,Tmax) Z Ω uq(·; t) ≤ C2: This, togetherwiththesecondequationandellipticregularitytheoryimplythat sup t∈(0,Tmax) Z Ω |∇v(·; t)|2q ≤ C3; for some C3 = C3(q) > 0. ByapplyingMoseriterationprocedureasin[1], [2] and[53], weobtain that sup t∈(0,Tmax) ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) < ∞: This, combinedwith(4.4.7), impliesthat Tmax = ∞ and uniformboundednessof (u; v). 66 4.5 Blow-uppreventionbysub-logisticsourcesin2dchemotaxissystemundernonlinear Neumann boundaryconditions This sectiondealswithclassicalsolutionstothechemotaxissystemwithsub-logisticsources, ru− μu2 lnp(u+e) , where r;p ≥ 0 and  > 0 under nonlinearNeumannboundaryconditioninasmooth bounded domain Ω ⊂ R2. Itisshownthatif p < 1 in fullyparabolicsystemsand p ≤ 1 in parabolic- ellipticsystems,thensolutionsexistgloballyandremainboundedintime.Ourproof relies onseveraltechniques,includingparabolicregularityinSobolevspaces,variationalargu- ments, interpolationinequalitiesinSobolevspaces,TraceSobolevembeddingtheoremandMoser iteration method. 4.5.1 Introduction In thissection,weconsiderthefollowingchemotaxismodelwithsub-logisticsourcesina bounded domainwithsmoothboundary Ω ⊂ R2: 8>>< >>: ut = Δu − ∇· (u∇v) + ru − μu2 lnp(u+e) vt = Δv − v + u; (4.5.1) where r;q ≥ 0,  > 0. Thesystem(4.5.1) iscomplementedwithnonnegativeinitialconditionsin C2+γ(Ω), where ∈ (0; 1), notidenticallyzero: u(x; 0) = u0(x); v(x; 0) = v0(x); with x ∈ Ω; (4.5.2) and nonlinearNeumannboundaryconditionareimposedasfollows: @u @ = g(u); @v @ = 0; x ∈ @Ω; t ∈ (0; Tmax); (4.5.3) where  is theoutwardnormalvectorand g is nonnegativein C1([0;∞]). The problem(4.5.1) withnonlinearboundarycondition(4.5.3) introducedandstudiedin[33] indicates thatthequadraticdegradationtermcanpreventblow-upinasmoothconvexbounded domain Ω ⊂ Rn with n ≥ 2 when g(s) = sq for q ≥ 1. Tomorespecific,if q ∈ 􀀀 1; 3 2  then solu- tions existgloballyandremainboundedwhen  > n−2 n , with n ≥ 2 and theborderlinecasewhen 67  = n−2 n , and p ∈ 􀀀 1; 1 + 1 n  with n ≥ 3 for parabolic-ellipticchemotaxissystem.Moreover,sim- ilar resultwasalsoobtainforfullyparabolicsystemwhen n = 2; 3. Especially,intwo-dimensional domain forany  > 0 and q ∈ 􀀀 1; 3 2  , thesystem(4.5.1) with p = 0 possesses auniquepositive classical solutionwhichremainsboundedatalltime.Therefore,itisnaturaltoask: Main Question: ”Can sub-logisticsourcesstillavoidblow-upinanonlinearNeumannbound- ary condition?” In thissection,ourobjectiveistoaddressthisquestionbyemployingmodifiedargumentsfrom[33] to handlethenonlineartermanddrawingupontechniquesfrom[71, 72] tohandlethesub-logistic sources. Wesummarizeourfindingsasfollows: Theorem4.5.1. Assume that (u; v) is alocalclassicalsolutionofthesystem (4.5.1) under the conditions (4.5.2), and (4.5.3) in Ω × (0; Tmax). If g satisfies thefollowingconditions: lim s→∞ |g′(s)| √ s lnp+1 2 (s + e) =0; with p < 1;  = 1; (4.5.4) or lim s→∞ |g′(s)| ln(s + e) ln1 √ 2 (ln(s + e)) s = 0; with p = 1;  = 0; (4.5.5) then Tmax = ∞ and (u; v) remainsboundedatalltimeinthesensethat sup t∈(0,∞) n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o < ∞: (4.5.6) Remark 4.5.1. The mainresultsin[33] isaspecialcaseofTheorem 4.5.1 when replacing p = 0, and g(s) = |s|q for 1 < q< 3 2 by condition (4.5.4) for fullyparabolicsystemsor (4.5.5) for parabolic-elliptic systems. Remark 4.5.2. Our analysisdoesnotworkwhenreplacingcondition (4.5.4) by aweakerone lim sup s→∞ |g′(s)| √ s lnp+1 2 (s + e) < ∞: As aimmediateconsequence,wehavethefollowingresult 68 Corollary4.5.2. Assume that p < 1, and (u; v) is alocalclassicalsolutionofthesystem (4.5.1) under theconditions (4.5.2), and (4.5.3) in Ω × (0; Tmax). If g satisfies g(s) = sq; or g(s) = s 3 2 lnk(s + e) for all s ≥ 0; (4.5.7) for any 1 < q< 3 2 and k > p+1 2 , then Tmax = ∞and (u; v) remainsboundedatalltimeinthesense that sup t∈(0,∞) n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o < ∞: (4.5.8) Remark 4.5.3. Weleavetheopenquestionwhetherthelogisticsourcescanstillpreventblow-up for g(u) = u 3 2 for  sufficiently small. Remark 4.5.4. One canalsoadoptandmodifytheproofofLemma 4.5.3 to obtaintheglobal boundedness resultforparabolic-ellipticcasewhen g(s) =  s 3 2 lnp+1 2 (s + e) for all s ≥ 0; for  > 0 sufficiently small. The sectionisorganizedinthreesubsections.Thekeyestimates,comprising L ln L and L2 estimates, areprovidedinSubsection 4.5.2. Finally,weintroducetheMoseriterationprocedure to obtainan L∞ bound forthesolution,andthenapplyittoprovethemainresultsinSubsection 4.5.3. Letusintroducelocalwellposednessresults,whichwasestablishedin[33]. Proposition4.5.1. If nonnegativefunctions u0; v0 arein C2+γ(¯Ω) such that @u0 @ = |u0|1+γ on @Ω; (4.5.9) where ∈ (0; 1). Thenthereexists Tmax ∈ (0;∞] such thatproblem (4.5.1) admits aunique nonnegative solution u; v in C2+γ,1+γ/2(¯Ω × (0; Tmax)). Moreover,if u0; v0 arenotidenticallyzero in Ω then u; v arestrictlypositivein Ω × (0; Tmax). If Tmax < ∞, then lim sup t→Tmax ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) = ∞: (4.5.10) 69 4.5.2 Aprioriestimates In thissection,wedenote a := ee and ”c” asauniversalconstantthatcanvarydependingon differentparametersandmaychangeovertime.Wealsoassumethat (u; v) is alocalclassical solution ofthesystem(4.5.1) undertheconditions(4.5.2), and(4.5.3) in Ω × (0; Tmax). Ouraim is toestablishaprioriestimateforthesolutions.Whilethemethodin[47] and[71] reliesonthe L1-estimate of u and theabsorptionof − R Ω |∇u 1 2 |2 to obtaina R Ω u ln u, wetakeadvantageofboth terms− R Ω |∇u 1 2 |2 and − u2 ln(u+e) . Letusbeginwithanestimateof R Ω u ln(u + e) as follows: Lemma 4.5.3. If thenthereexists C > 0 such thatforall t ∈ (0; Tmax), wehave Z Ω u(·; t) ln(u(·; t) + e) + |∇v(·; t)|2 < C: (4.5.11) Proof. Let call y(t) := Z Ω u ln(u + e) + 1 2 |∇v|2; we have y′(t) = Z Ω  ln(u + e) + u u + e  Δu − ∇· (u∇v) + ru − u2 lnp(u + e)  + Z Ω ∇v · ∇(Δv − v + u) := I + J: (4.5.12) By integrationbyparts, I can berewrittenas: I = − Z Ω |∇u|2 u + e − Z Ω e|∇u|2 (u + e)2 + Z Ω  u u + e + eu (u + e)2  ∇u · ∇v + Z Ω  ln(u + e) + u u + e  ru − u2 lnp(u + e)  + Z ∂Ω  ln(u + e) + u u + e  g(u) dS: (4.5.13) Let denote (u) := Z u 0 s s + e + es (s + e)2 ds 70 we seethat (u) ≤ u. Byintegrationbypartsandelementaryinequality,wehavethatforany  > 0 there exists c > 0 depending on  such that Z Ω  u u + e + eu (u + e)2  ∇u · ∇v = Z Ω ∇(u) · ∇v = − Z Ω (u)Δv ≤  Z Ω (Δv)2 + 1 4 Z Ω 2(u) ≤  Z Ω (Δv)2 + 1 4 Z Ω u2 ≤  Z Ω (Δv)2 +  Z Ω u2 ln1−p(u + e) + c; (4.5.14) where thelastinequalitycomesfromthefactthatforany  > 0, thereexists C > 0 depending on  such that u2 ≤ u2 ln1−p(u + e) + C(); 0 ≤ p < 1: This alsoimpliesthatforany  > 0, thereexists c > 0 depending on  such that Z Ω  ln(u + e) + u u + e  ru − u2 lnp(u + e)  ≤ ( − ) Z Ω u2 ln1−p(u + e) + c: (4.5.15) By traceSobolev’sembeddingtheorem W1,1(Ω) → L1(@Ω), wehave Z ∂Ω  ln(u + e) + u u + e  g(u) dS ≤ C Z Ω  ln(u + e) + u u + e  |g(u)| + C Z Ω  1 u + e + e (u + e)2  |g(u)||∇u| + C Z Ω  ln(u + e) + u u + e  |g′(u)||∇u|: (4.5.16) The conditions(4.5.4) impliesthat lim s→∞ |g(s)| s 3 2 = 0: This, togetherwithelementaryinequalitiesdeducesthatforany  > 0, thereexists c > 0 depending on  such that C Z Ω  ln(u + e) + u u + e  |g(u)| ≤  Z Ω u2 ln1−p(u + e) + c: (4.5.17) 71 By similarargument,onecanalsoverifythat C Z Ω  1 u + e + e (u + e)2  |g(u)||∇u| ≤ 2C Z Ω 1 u + e |g(u)||∇u| ≤  Z Ω |∇u|2 u + e + C2  Z Ω g2(u) u + e ≤  Z Ω |∇u|2 u + e +  Z Ω u2 ln1−p(u + e) + c; (4.5.18) where c > 0 depending on . Fromthecondition(4.5.4), wehave C Z Ω  ln(u + e) + u u + e  |g′(u)||∇u| ≤ 2C Z Ω ln(u + e)|g′(u)||∇u| ≤  Z Ω |∇u|2 u + e + C2  Z Ω (u + e) ln2(u + e)|g′(u)|2 ≤  Z Ω |∇u|2 u + e +  Z Ω u2 ln1−p(u + e) + c; (4.5.19) for any  > 0, and c = c() > 0. Collectingfrom(4.5.16) to(4.5.19) andreplacing  by /3, we have Z ∂Ω  ln(u + e) + u u + e  g(u) dS ≤  Z Ω |∇u|2 u + e +  Z Ω u2 ln1−p(u + e) + c; (4.5.20) for any  > 0, and c = c() > 0. Combining(4.5.13), (4.5.14), (4.5.15), and(4.5.20), andchoosing  sufficientlysmall,weobtain I ≤ −1 2 Z Ω |∇u|2 u + e −  2 Z Ω u2 ln1−p(u + e) +  Z Ω (Δv)2 + c: (4.5.21) By integrationbypartsandelementalinequalities,weobtainthatforany  > 0, thereexist c > 0 depending on  such that J = − Z Ω (Δv)2 − Z Ω |∇v|2 − Z Ω uΔv ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 + 1 2 Z Ω u2 ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 +  Z Ω u2 ln1−p(u + e) + c: (4.5.22) For any  > 0, thereexistapositiveconstant c > 0 depending on  such that Z Ω u ln(u + e) ≤  Z Ω u2 ln1−p(u + e) + c: (4.5.23) 72 This, togetherwith(4.5.12), (4.5.21), and(4.5.22), impliesthat y′(t) + y(t) ≤ ( − 1 2 ) Z Ω |∇u|2 − 1 2 Z Ω |∇v|2 + ( −  2 ) Z Ω u2 ln1−p(u + e) + c: (4.5.24) Wechoose  sufficientlysmallandapplyGronwall’sinequalitytocompletetheproof. The followinglemmaprovidesusanessentialestimateontheboundary.Thefollowinglemma provides usanessentialestimateontheboundary. Lemma 4.5.4. Assume that g satisfies thecondition (4.5.5), thenforany  > 0, thereexistsa positive constant C depending on  such thatforany u ∈ C1(¯Ω): Z ∂Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  g(u) ≤  Z Ω |∇u|2 (u + a) ln(u + a) +  Z Ω u2 ln(ln(u + a)) ln(u + a) + C: (4.5.25) Proof. By traceSobolevembeddingtheorem W1,1(Ω) → L1(@Ω), wehave Z ∂Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  g(u) ≤ C Z Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  g(u) + C Z Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  |g′(u)||∇u| + C Z Ω  1 (u + a) ln(u + a) + (u + a) ln(u + a) − u ln(u + a) − u (u + a)2 ln2(u + a)  |∇u||g(u)| := I + J + K: (4.5.26) The condition(4.5.5) entailsthatforany  > 0 I ≤  Z Ω u2 ln(ln(u + a)) ln(u + a) + c(): (4.5.27) By applyingYoung’sinequalityandthenusingcondition(4.5.5), wehave J ≤  Z Ω |∇u|2 (u + a) ln(u + a) + c() Z Ω (u + a) ln(u + a) ln(ln(u + a))|g′(u)|2 ≤  Z Ω |∇u|2 (u + a) ln(u + a) +  Z Ω u2 ln(ln(u + a)) ln(u + a) + c(): (4.5.28) One canverifythat K ≤ C Z Ω 1 (u + a) ln(u + a) |∇u||g(u)|: 73 Applying Young’sinequalitytotheright,weobtain K ≤  Z Ω |∇u|2 (u + a) ln(u + a) + c() Z Ω |g(u)|2 (u + a) ln(u + a) ≤  Z Ω |∇u|2 (u + a) ln(u + a) +  Z Ω u2 ln(ln(u + a)) ln(u + a) + c(); (4.5.29) where thelastinequalitycomesfromaconsequenceofcondition(4.5.5): lim s→∞ |g(s)| s3/2 = 0: Collecting from(4.5.26) to(4.5.29), weobtain(4.5.25). Next, wederiveaprioriestimateforsolutionsofparabolic-ellipticsystems. Lemma 4.5.5. If  = 0 and p = 1 then thereexists C > 0 such thatforall t ∈ (0; Tmax), wehave Z Ω u(·; t) ln(ln(u(·; t) + a)) ≤ C: (4.5.30) Proof. Wedefine y(t) = R Ω u ln(ln(u + a)) and differentiate y to obtain y′(t) = Z Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  ut = Z Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  Δu − ∇· (u∇v) + ru −  u2 ln(u + a)  = − Z Ω ∇  ln(ln(u + a)) + u (u + a) ln(u + a)  · ∇u + Z Ω u∇  ln(ln(u + a)) + u (u + a) ln(u + a)  · ∇v + Z Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  ru −  u2 ln(u + a)  + Z ∂Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  g(u) := I + J + K + L (4.5.31) By integrationbyparts,wehave I = − Z Ω ∇  ln(ln(u + a)) + u (u + a) ln(u + a)  · ∇u = − Z Ω  1 (u + a) ln(u + a) + a ln(u + a) − u (u + a)2 ln2(u + a)  |∇u|2 74 = − Z Ω u ln(u + a) +2a ln(u + a) − u (u + a)2 ln2(u + a) |∇u|2 ≤ −1 2 Z Ω |∇u|2 (u + a) ln(u + a) : (4.5.32) Similarly,wehave J = Z Ω u∇  ln(ln(u + a)) + u (u + a) ln(u + a)  · ∇v = Z Ω u2(ln(u + a) − 1) +2au ln(u + a) (u + a)2 ln2(u + a) ∇u · ∇v = Z Ω ∇(u) · ∇v = Z Ω (u)(u − v) ≤ Z Ω u(u); (4.5.33) where 0 ≤ (u) := Z u 0 s2(ln(s + a) − 1) +2as ln(s + a) (s + a)2 ln2(s + a) ds ≤ Z u 0 1 ln(s + a) ds: (4.5.34) Thus, weobtain J ≤ Z Ω u Z u 0 1 ln(s + a) ds: (4.5.35) By L’Hospitallemma,wehave lim u→∞ R u 0 1 ln(s+a) ds u ln(ln(u+a)) ln(u+a) = lim u→∞ ln(u + a) ln(u + a) ln(ln(u + a)) + u u+a − u u+a ln(ln(u + a)) = 0: (4.5.36) Therefore, forany  > 0, thereexist N depending on  such thatfor u >N, wehave Z u 0 1 ln(s + a) ds ≤ u ln(ln(u + a)) ln(u + a) : (4.5.37) This leadsto Z Ω u Z u 0 1 ln(s + a) ds = Z u≤N u Z u 0 1 ln(s + a) ds + Z u>N u Z u 0 1 ln(s + a) ds ≤  Z Ω u2 ln(ln(u + a)) ln(u + a) + c (4.5.38) where c = N2|Ω|. From(4.5.35) and(4.5.38), wehave J ≤  Z Ω u2 ln(ln(u + a)) ln(u + a) + c: (4.5.39) 75 One canverifythatforany  > 0, thereexist C() > 0 such that K = Z Ω  ln(ln(u + a)) + u (u + a) ln(u + a)  ru −  u2 ln(u + a)  ≤ ( − ) Z Ω u2 ln(ln(u + a)) ln(u + a) + c(): (4.5.40) From (4.5.25), wehave L ≤  Z Ω |∇u|2 (u + a) ln(u + a) +  Z Ω u2 ln(ln(u + a)) ln(u + a) + c(): (4.5.41) Furthermore, wehave y(t) ≤  Z Ω u2 ln(ln(u + a)) ln(u + a) + c: (4.5.42) Collect (4.5.31), (4.5.32), (4.5.35), (4.5.39),(4.5.40), (4.5.41) and(4.5.42), wehave y′(t) + y(t) ≤ (4 − ) Z Ω u2 ln(ln(u + a)) ln(u + a) + c: (4.5.43) Wechoose  sufficientlysmallandapplyGronwall’sinequalitytoimply y(t) ≤ C for all t > 0. Consequently,an L2-estimate of u is derivedasfollows: Lemma 4.5.6. If g satisfies (4.5.4) and inequality (4.5.11) holds thenthereexists C > 0 such that Z Ω u2(·; t) +  Z Ω |Δv(·; t)|2 < C; (4.5.44) for all t ∈ (0; Tmax). Proof. Let call y(t) = 1 2 Z Ω u2 +  2 Z Ω (Δv)2; we have y′(t) = Z Ω u  Δu − ∇· (u∇v) + ru − u2 lnp(u + e)  +  Z Ω ΔvΔvt 76 = I + J: (4.5.45) By integrationbyparts, I can berewrittenas I = − Z Ω |∇u|2 + Z Ω u∇u · ∇v + r Z Ω u2 −  Z Ω u3 lnp(u + e) + Z ∂Ω ug(u) dS: (4.5.46) In case  = 0, weusethesecondequationof(4.5.1) toobtain Z Ω u∇u · ∇v = − Z Ω u2Δv = Z Ω u2(u − v) ≤ Z Ω u3: (4.5.47) When  = 1, weuseYoung’sinequalitytoobtain Z Ω u∇u · ∇v = − Z Ω u2Δv ≤  Z Ω (Δv)3 + c Z Ω u3; (4.5.48) where c > 0 depending on . BytraceSobolev’sembeddingTheorem W1,1(Ω) → L1(@Ω), we obtain Z ∂Ω ug(u) dS ≤ C Z Ω u|g(u)| + C Z Ω |g(u)||∇u| + C Z Ω u|g′(u)||∇u|: When g satisfies condition(4.5.4), weapplyYoung’sinequalitytothelasttwotermsoftheright hand sidetoobtain Z ∂Ω ug(u) dS ≤  Z Ω |∇u|2 +  Z Ω u3 + c; (4.5.49) where c > 0 depending on . Wemakeuseofthefollowinginequalityestablishedin[30][Lemma 3.3] forany  > 0 Z Ω u3 ≤  Z Ω |∇u|2 Z Ω G(u) + C Z Ω u 3 + c() Z Ω u  ; where G(u) = 8>>< >>: u ln(u + e); when  = 1 u ln(ln(u + a)); when  = 0: WeapplyLemma 4.5.3 to obtainthat Z Ω u3 ≤  Z Ω |∇u|2 + c; (4.5.50) 77 where c > 0 depending on . Collecting(4.5.46), (4.5.48), (4.5.49)and (4.5.50), wehave I ≤ ( − 1) Z Ω |∇u|2 +  Z Ω (Δv)3 + c; for any  > 0, and c > 0 depending on . Sincesupt∈(0,Tmax) R Ω |∇v|2 < ∞, thefollowinginequality ( see[68]) holds Z Ω (Δv)3 ≤ C Z Ω |∇Δv|2 + C: This leadsto I ≤ ( − 1) Z Ω |∇u|2 +  Z Ω |∇Δv|2 + c; (4.5.51) for any  > 0, and c > 0 depending on . Byintegrationbyparts,andYoung’sinequality,wehave J = − Z Ω |∇Δv|2 −  Z Ω (Δv)2 −  Z Ω ∇Δv · ∇u ≤ − 2 Z Ω |∇Δv|2 −  Z Ω (Δv)2 +  2 Z Ω |∇u|2: (4.5.52) Collecting (4.5.51),(4.5.52), andusingthefollowinginequality Z Ω u2 ≤  Z Ω |∇u|2 + c(); thereafter choosing  sufficientlysmall,weobtainthat y′(t) + y(t) ≤ C; (4.5.53) for somepositiveconstant C. This,togetherwithGronwall’sinequalityassertsthat y(t) ≤ max {y(0);C} for all t ∈ (0; Tmax). 4.5.3 Regularityandproofofmainresults In thissection,weshowthatif u is uniformlyboundedintimeunder ∥·∥ L2(Ω) then itisalso uniformly boundedintimeunder ∥·∥ L∞(Ω). Weconsiderthefollowingequation,whichismore general than(4.5.1) 8>>< >>: ut = Δu − ∇· (u∇v) + f(u) vt = Δv − v + u; (4.5.54) where f is continuoussuchthat f(s) ≤ c + cs2 for all s ≥ 0 with c ≥ 0. 78 Theorem4.5.7. Assume that g(s) ≤ s 3 2 ; s ≥ 0; (4.5.55) where > 0. Let (u; v) be aclassicalsolutionof (4.5.54) under conditions (4.5.2) and (4.5.3) in Ω × (0; Tmax) with maximalexistencetime Tmax ∈ (0;∞]. If sup t∈(0,Tmax) ∥u(·; t)∥ L2(Ω) < ∞; then sup t∈(0,Tmax)  ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω)  < ∞: ProofofTheorem 2.5.1. It istheimmediateconsequenceofTheorem C.2.1 when n = 2, g(u) = u 3 2 and f(u) = ru − μu2 lnq(u+e) ≤ ru. Wearenowreadytoprovethemaintheorem ProofofTheorem 4.5.1. It istheimmediateconsequenceofLemma 4.5.3, 4.5.5, 4.5.6 and Theo- rem 4.5.7. 79 CHAPTER 5 SUPERLINEAR CROSS-DIFFUSION;SUPERLINEARSIGNALPRODUCTION In thischapter,weinvestigatetheglobalexistenceofsolutionstosomechemotaxismodelswith superlinear crossdiffusionratesandsuperlinearsignalproduction.Itisshownthattheappreanceof the quadraticdegradationtermscanensuretoexcludeblow-upphenomenoninthosemodels.The pivotal analysistoolistheregularitytheoryinOrliczspacesforellipticandparabolicequations, which enablesustoeliminatedegeneraciesofthediffusionterms.Subsequently,wecanapplythe well-established frameworkinpreviouschapterstoobtaintheglobalexistenceandboundednessof solutions. 5.1 Degeneratechemotaxissystemswithsuperlineargrowthincross-diffusionratesandlo- gistic sources The objectiveistoinvestigatetheglobalexistenceofsolutionsfordegeneratechemotaxissys- tems withlogisticsourcesinatwo-dimensionaldomain.Itisdemonstratedthattheinclusionof logistic sourcescanexcludetheoccurrenceofblow-upsolutions,eveninthepresenceofsuperlin- ear growthinthecross-diffusionrate.Ourproofreliesontheapplicationofellipticandparabolic regularity inOrliczspacesandvariationalapproach. 5.1.1 Introduction Weconsiderthefollowingsystemarisingfromchemotaxisinasmoothboundeddomain Ω ⊂ R2: 8>>< >>: ut = ∇ · (D(v)∇(u)) − ∇· (S(v)u lnα(u + e)∇v) + ru − u2 vt = Δv − v + u; (5.1.1) where  ∈ {0; 1}, r ∈ R,  ≥ 0, ≥ 0, and 0 < D ∈ C2([0;∞)) and S ∈ C2([0;∞)) ∩W1,∞((0;∞)) such that S′ ≥ 0: (5.1.2) The system(5.1.1) iscomplementedwithnonnegative,initialconditionsinW1,∞(Ω) not identically zero: u(x; 0) = u0(x); v(x; 0) = v0(x); with x ∈ Ω; (5.1.3) 80 and homogeneousNeumannboundaryconditionareimposedasfollows: @u @ = @v @ = 0; x ∈ @Ω; t ∈ (0; Tmax); (5.1.4) where  denotes theoutwardnormalvector. Our maingoalistoshowthatthequadraticlogisticdegradationtermcaneffectivelypreventblow- up forbothelliptic-parabolicandfullyparabolicdegeneratechemotaxismodelswithsuperlinear growth inthecross-diffusionratewhere > 0. Tobemoreprecise,ourmainresultreadsas follows: Theorem5.1.1. Suppose that  = 0 and ∈ (0; 1) then thesystem (5.1.1) under theassumptions (5.1.2), (5.1.3) and (5.1.4) admits aglobalclassicalsolution (u; v) in  C0 􀀀 ¯Ω × [0;∞)  ∩ C2,1 􀀀 ¯Ω × (0;∞) 2 such that u > 0 and v > 0 in ¯Ω × (0;∞). Furthermore, this solutionisboundedinthesensethat sup t>0 n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o < ∞: (5.1.5) For fullyparaboliccases,wehavethefollowingtheorem: Theorem5.1.2. Suppose  = 1 and ∈ (0; 1 2 ) then thesystem (5.1.1) under theassumptions (5.1.2), (5.1.3) and (5.1.4) admits aglobalclassicalsolution (u; v) with 8>>< >>: u ∈ C0 􀀀 ¯Ω × [0;∞)  ∩ C2,1 􀀀 ¯Ω × (0;∞)  and v ∈ ∩q>2C0 ([0;∞);W1,q(Ω)) ∩ C2,1 􀀀 ¯Ω × (0;∞)  ; such that u > 0 and v > 0 in ¯Ω × (0;∞). Furthermore,thissolutionisboundedinthesensethat sup t>0 n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o < ∞: (5.1.6) The proofofthemainresultscanbesummarizedintothreesteps: 1. Derive aninitialestimateforsolutions: sup t∈(0,Tmax) Z Ω u lnk(u + e) + |∇v|2 < ∞; for some k ≥ 1: 81 Toaccomplishthis,weadaptandmodifytheargumentpresentedin[30][Lemma 4.1]forthe proof ofLemma 5.1.7 and 5.1.10. 2. Addressthedegeneracyofthediffusionterm: Eliminate thedegeneracyofthediffusion term byemployingellipticandparabolicregularityinOrliczspaces.Theproofoftheelliptic part isprovidedinLemma 5.1.6, andweapplytheparabolicpartasestablishedin[61]. 3. Establish Lp bounds forthesolution: Lemma 5.1.8 and 5.1.11 establish Lp bounds forthe solution forany p > 1. Theprimarychallengeliesinincorporatingtheterm R Ω up lnα(u+e) into thediffusionterm.Overcomingthisdifficultyinvolvestheutilizationoflogarithmically refined Gagliardo-Nirenberginterpolationinequalities,asestablishedin[66]. This sectionisstructuredasfollows.InSubsection 5.1.2, werevisitlocalexistenceresultsfor both elliptic-parabolicandfullyparabolicmodels,alongwithkeyinequalitiesusedinsubsequent sections. WealsoprovideresultsonregularityinOrliczspaces.Subsection 5.1.3 presents apriori estimates, including L lnk L and Lp estimates forsolutionsofelliptic-parabolicmodelswhen  = 0, andincludestheproofofTheorem 5.1.1. Subsection 5.1.4 follows asimilarframeworkbut addresses thefullyparaboliccasewhen  = 1 and includestheproofofTheorem 5.1.2. 5.1.2 Preliminaries By employingfixedpointargumentsandapplyingstandardtheoriesofellipticandparabolic regularity,wecanestablishthelocalexistenceanduniquenessofnon-negativeclassicalsolutions to thesystem(5.1.1). Ourinitialstepinvolvesestablishingthelocalexistenceofsolutionsfor parabolic-elliptic chemotaxismodels,andweachievethisbyadaptingthemethodpresentedin [59][Theorem 2.1]. Lemma 5.1.3. Let  = 0 and Ω ⊂ R2 be aboundeddomainwithsmoothboundaryandthat (5.1.2), (5.1.3), and (5.1.4) hold. Thenthereexist Tmax ∈ (0;∞] and functions (u; v) in  C0 􀀀 ¯Ω × [0; Tmax)  ∩ C2,1 􀀀 ¯Ω × (0; Tmax) 2 such that u > 0 and v > 0 in ¯Ω × (0;∞), that (u; v) 82 solves (5.1.1) classically in Ω × (0; Tmax), andthat if Tmax < ∞; then lim sup t→Tmax n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o = ∞: (5.1.7) The localexistenceofsolutionsforfullyparabolicmodelscanbeattainedbymodifyingand adjusting theproofin[64][Lemma 1.1]orreferringto[21, 29]. Lemma 5.1.4. Let  = 1 and Ω ⊂ R2 be aboundeddomainwithsmoothboundaryandthat (5.1.2), (5.1.3), and (5.1.4) hold. Thenthereexist Tmax ∈ (0;∞] and functions 8>>< >>: u ∈ C0 􀀀 [0; Tmax);C0(¯Ω)  ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  and v ∈ T q>2 C0 ([0; Tmax);W1,q(Ω)) ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  (5.1.8) such that u > 0 and v > 0 in ¯Ω ×(0;∞), that (u; v) solves (5.1.1) classically in Ω×(0; Tmax), and that if Tmax < ∞; then lim sup t→Tmax n ∥u(·; t)∥ L∞(Ω) + ∥u(·; t)∥ W1,∞(Ω) o = ∞: (5.1.9) The followingLemma[56][Lemma A.1]providesausefulpointwiseestimateforGreen’sfunc- tion of −Δ +1. Lemma 5.1.5. Suppose that Ω ⊂ R2 is aboundeddomainwithsmoothboundary,andlet G denote Green’sfunctionof −Δ +1 in Ω subject toNeumannboundaryconditions.Thenthereexist A > diam(Ω) and K > 0 such that |G(x; y)| ≤ K ln A |x − y| for all x; y ∈ Ω with x ̸= y: (5.1.10) By thepointwiseestimatefortheGreen’sfunctionandLegendretransform,wecanderivea L∞ bound forsolutionsof(5.1.1) when  = 0, andthereforeeliminatethedegeneracyofdiffusion term. Lemma 5.1.6. Let Ω ⊂ R2 be aboundeddomainwithsmoothboundary.Supposethatthenon- negative function f in L2(Ω) satisfies Z Ω f ln(f + e) ≤ M (5.1.11) 83 and w is asolutionsof 8>>< >>: −Δw + w = f;x ∈ Ω ∂u ∂ν = 0; x ∈ @Ω; (5.1.12) then wehave ∥w∥ L∞(Ω) ≤ C; where C = C(M) > 0 Proof. By usingtheGreen’sfunction G of −Δ +1 in Ω, Lemma 5.1.5 and theinequalitythat ab ≤ a ln a + eb−1; for all a; b ≥ 0; we deducethat w(x) = Z Ω G(x; y)f(y) dy ≤ K Z Ω ln A |x − y|f(y) dy ≤ K Z Ω f(y) ln f(y) + K Z Ω eln A |x−y| −1 ≤ KM + AK e Z Ω 1 |x − y| dy ≤ KM + AK e diam(Ω): (5.1.13) 5.1.3 Elliptic-Parabolicsystem Let usbeginthissectionwithan L lnk L estimate forsolutionsof(5.1.1). Thekeyapproach in theproofisgroundedintheLyapunovfunctionalmethod.Whileastandardestimateintwo- dimensional domainsisoftenconsideredwhen k = 1, weaimtoenhanceitbyexploringthecase where k ≥ 1. TheinspirationisdrawnfromtheconstructionofaLyapunovfunctionalinan unconventional manner,asintroducedin[72]. Thisideahasbeenadaptedandrefinedin[30] for addressing two-dimensionalchemotaxismodelswithadegeneratediffusionterm,andin[32] for 84 two-species withtwochemicals,althoughthelogisticsourceappearsonlyinoneofthetwodensity population equations. Lemma 5.1.7. Under theassumptionsinTheorem 5.1.1, forany k ≥ 1, wehavethat sup t∈(0,Tmax) Z Ω u(·; t) lnk(u(·; t) + e) < ∞: (5.1.14) Proof. Wedefine I(t) := Z Ω u lnk(u + e) and differentiate I(·) to obtain I′(t) = Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  (∇ · (D(v)∇u − uS(v) lnα(u + e)∇v) + f(u)) = −k Z Ω D(v) lnk−1(u + e) u + e |∇u|2 − k(k − 1) Z Ω D(v)u lnk−2(u + e) (u + e)2 |∇u|2 − k Z Ω eD(v) lnk−1(u + e) (u + e)2 |∇u|2 + Z Ω S(v)∇(u) · ∇v + Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  (ru − u2) ≤ Z Ω S(v)∇(u) · ∇v + Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  (ru − u2) (5.1.15) where (l) := Z l 0  ks lnk+α−1(s + e) s + e + k(k − 1)u2 lnk+α−2(s + e) (s + e)2 + kes lnk+α−1 (s + e)2  ds ≤ c1l lnk+α−1(l + e); for all l ≥ 0; (5.1.16) with c1 = k2 + k. Byusingintegrationbyparts,takingintoaccountthecondition S′ ≥ 0 and applying elementaryinequalities,weobtainthat Z Ω S(v)∇(u) · ∇v = − Z Ω S(v)(u)Δv − Z Ω S′(v)(u)|∇v|2 ≤ ∥S∥ L∞((0,∞)) Z Ω (u)u ≤ c1 ∥S∥ L∞((0,∞)) Z Ω u2 lnk+α−1(u + e) ≤  4 Z Ω u2 lnk(u + e) + c2; (5.1.17) 85 where c2 = C(; ;k) > 0 and thelastinequalitycomesfromthefactthat k + − 1 < k when < 1 and theinequalitythatforany  > 0, thereexist A = c() > 0 such that sa1 lnb1(s + e) ≤ sa2 lnb2(s + e) + A; for all s ≥ 0; (5.1.18) where a1; a2; b1; b2 are positivenumberssuchthat a1 < a2. Tohandlethelasttermof(5.1.15) ,we make useofagain(5.1.18) toobtain Z Ω  lnk (u + e) + k lnk−1 (u + e) u + e  (ru − u2) ≤ r Z Ω u lnk(u + e) + r Z Ω k lnk−1(u + e) −  Z Ω u2 lnk(u + e) ≤ − 4 Z Ω u2 lnk(u + e) + c3; (5.1.19) where c3 = C() > 0 .The inequality(5.1.18) alsoimpliesthatthereexists c4 = C() > 0 such that Z Ω u lnk(u + e) ≤  4 Z Ω u2 lnk(u + e) + c4: (5.1.20) Collecting (5.1.15), (5.1.17), (5.1.19), and(5.1.20) yields I′(t) + I(t) ≤ c5; (5.1.21) where c5 = c2 + c3 + c4. Finally,weapplyGronwall’sinequalitytoprove(5.1.14). Wewillestablishan Lp estimate forthesolutioninthefollowinglemma.Whenemployingthe standard testingapproachcommonlyusedinchemotaxis,controllingtheterm R Ω up+1 lnα(u + e) proves challengingusingthediffusionterm − R Ω |∇u p 2 |2 and theglobalboundednessof R Ω u. To overcome thisdifficulty,thekeyideaistoutilizethebound R Ω u lnk(u+e) instead of R Ω u and the logarithmically refinedGagliardo-NirenberginterpolationinequalityinLemma B.0.5. Lemma 5.1.8. Under theassumptionsinTheorem 5.1.1, forany p > 1, wehavethat sup t∈(0,Tmax) Z Ω up(x; t)dx < ∞: (5.1.22) 86 Proof. By integrationbyparts,wehave 1 p d dt Z Ω up = Z Ω up−1 􀀀 ∇(D(v)∇u) − ∇· (S(v)u lnα(u + e)∇v) + ru − u2 = −2(p − 1) p Z Ω D(v)|∇u p 2 |2 + (p − 1) Z Ω S(v)up−1 lnα(u + e)∇u · ∇v + r Z Ω up −  Z Ω up+1 (5.1.23) From Lemma 5.1.7, thereexistaconstant M > 0 such that Z Ω u(·; t) ln(u(·; t) + e) ≤ M; for all t ∈ (0; Tmax) (5.1.24) This, togetherwithLemma 5.1.6 implies that ∥v(·; t)∥ L∞(Ω) ≤ C; for all t ∈ (0; Tmax) (5.1.25) for some C = C(M) > 0. Thisimpliesthatinf(x,t) D(v(x; t) > 0 and thereforethedegeneracyof the diffusiontermisnoweliminated.Itfollowsthat −2(p − 1) p Z Ω D(v)|∇u p 2 |2 ≤ −c1 Z Ω |∇u p 2 |2; (5.1.26) where c1 = 2p−2 p inf(x,t)∈Ω×(0,T ) D(v(x; t)). Byintegrationbypartsandthecondition S′ ≥ 0, we have (p − 1) Z Ω S(v)up−1 lnα(u + e)∇u · ∇v = −c2 Z Ω S(v)(u)Δv − c2 Z Ω S′(v)(u)|∇v|2 ≤ c3 Z Ω u(u) ≤ c3 Z Ω up+1 lnα(u + e): (5.1.27) where (l) := Z l 0 sp−1 lnα(s + e) ds ≤ lp lnα(l + e); for all l ≥ 0: (5.1.28) From Lemma 5.1.7, weobtainthatsupt∈(0,Tmax) R Ω u ln(u + e) < ∞. NowapplyingLemma B.0.5 with  = c1 2c3 supt∈(0,Tmax) R Ω u ln(u + e) 87 yields c3 Z Ω up+1 lnα(u + e) ≤ c3 Z Ω |∇u p 2 |2 · Z Ω u ln(u + e) + c3 Z Ω u · Z Ω u ln(u + e) + c4 ≤ c1 2 Z Ω |∇u p 2 |2 + c5 (5.1.29) where c4 = C() > 0 and c5 = C() > 0. ByYoung’sinequality,weobtainthat  r + 1 p Z Ω up ≤  2 Z Ω up+1 + c6: (5.1.30) where c6 = C(r;p;) > 0. Collecting(5.1.23), (5.1.26), (5.1.27) and(5.1.30) yields 1 p d dt Z Ω up + 1 p Z Ω up ≤ c7; (5.1.31) where c7 = C(; p;;r) > 0. Finally,weapplyGronwall’sinequalitytocompletetheproof. Wearenowreadytoprovethemaintheorem. ProofofTheorem 5.1.1. By usingLemma 5.1.8 for afixed p > 2, itfollowsthat sup t∈(0,Tmax) ∥u(·; t)∥ Lp(Ω) < ∞: By ellipticregularitytheoryinSobolevspaces,weobtainthat sup t∈(0,Tmax) ∥v(·; t)∥ W1,∞(Ω) < ∞: (5.1.32) Applying Moser-Alikakositeration(seee.g[53, 2, 1] )yields sup t∈(0,Tmax) ∥u(·; t)∥ L∞(Ω) < ∞: This, togetherwithLemma 5.1.4 implies that Tmax = ∞, whichfinishestheproof. 5.1.4 FullyParabolicsystem WewillfollowtheframeworkestablishedintheprevioussectiontoproveTheorem 5.1.2. How- ever,forthefullyparabolicsystem,wecannotdirectlyusetheequationΔv = u−v for estimations, as doneinLemmas 5.1.7 and 5.1.8. Instead,weneedtoestablishanintermediateestimatetocon- nect thetwoequationsof(5.1.1). Letuscommencethissectionwiththefollowinglemma. 88 Lemma 5.1.9. For any p > 1, thereexistpositiveconstants A1;A2;A3 depending onlyon p such that 1 2p d dt Z Ω |∇v|2p + A1 Z Ω |∇|∇v|p|2 + Z Ω |∇v|2p ≤ A2 Z Ω u2|∇v|2p−2 + A3 Z Ω |∇v|2p (5.1.33) Proof. Wemakeuseofthefollowingpoint-wiseidentity ∇v · ∇Δv = 1 2 Δ(|∇v|2) − |D2v|2 to obtain 1 2p d dt Z Ω |∇v|2p + Z Ω |∇v|2p = −c1 Z Ω |∇|∇v|p|2 − Z Ω |∇v|2p−2|D2v|2 + Z Ω |∇v|2p−2∇v · ∇u + c2 Z ∂Ω @|∇v|2 @ |∇v|2p−2; (5.1.34) where c1; c2 are positiveconstantsdependingonlyon p. Theinequality ∂|∇v|2 ∂ν ≤ M|∇v|2; (see [41][Lemma 4.2])forsome M > 0 depending onlyon Ω, impliesthat c2 Z ∂Ω @|∇v|2 @ |∇v|2p−2 dS ≤ c2M Z ∂Ω |∇v|2p dS: Let g := |∇v|p and applytraceembeddingtheorem W1,1(Ω) −→ L1(@Ω) together withYoung’s inequality,thereexistpositiveconstants C and c3 such that c2M Z ∂Ω g2 dS ≤ C Z Ω g|∇g| + C Z Ω g2 ≤ c1 2 Z Ω |∇g|2 + c3 Z Ω g2; (5.1.35) Therefore, wehave c2M Z ∂Ω |∇v|2p dS ≤ c1 2 Z Ω |∇|∇v|p|2 + c3 Z Ω |∇v|2p: (5.1.36) Applying thepointwiseinequality (Δv)2 ≤ 2|D2v|2 to (5.1.34) yields 1 2p d dt Z Ω |∇v|2p + Z Ω |∇v|2p ≤ −c1 2 Z Ω |∇|∇v|p|2 − 1 2 Z Ω |∇v|2p−2|Δv|2 89 + Z Ω |∇v|2p−2∇v · ∇u + c3 Z Ω |∇v|2p (5.1.37) By integrationbypartsandelementalinequalities,thereexistconstants c4 = C(p) > 0 and c5 = C(p) > 0 in suchawaythat Z Ω |∇v|2p−2∇v · ∇u = − Z Ω u|∇v|2p−2Δv − c4 Z Ω u|∇v|p−1∇|∇v|p · ∇v |∇v| ≤ 1 2 Z Ω (Δv)2|∇v|2p−2 + c1 4 Z Ω |∇|∇v|p|2 + c5 Z Ω u2|∇v|2p−2; (5.1.38) From (5.1.37) and(5.1.38), wefinallyprove(5.1.33). The followinglemma,akintoLemma 5.1.7, providesacrucialaprioriestimateforsolu- tions. However,theconstant k is nowboundedfromaboveduetothestructureofparabolicequa- tions. Inadditiontotheestimatefor R Ω u lnk(u + e), wealsorequireauniformboundintimefor R t+τ t R Ω u2 lnk(u + e) to cooperatewithProposition C.1.1 in ordertoobtainuniformboundsfor v. Lemma 5.1.10. Under theassumptionsinTheorem 5.1.2, forany k ∈ (1; 2 − 2 ), wehavethat sup t∈(0,Tmax) Z Ω  u lnk (u + e) + |∇v|2 + sup t∈(0,Tmax−τ) Z t+τ t Z Ω u2 lnk(u + e) < ∞; (5.1.39) where  = min  1; Tmax 2 . Proof. Wedefine y(t) := Z Ω u lnk (u + e) + 1 2 Z Ω |∇v|2; and differentiate y(·) to obtain y′(t) = Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  ut + Z Ω ∇v · ∇vt := I′(t) + J′(t): (5.1.40) Where I is giveninLemma 5.1.7. Wenowjustreuseestimationsfrom(5.1.15) to(5.1.19) for I′ except for(5.1.17). Byusingintegrationbyparts,takingintoaccountthecondition S′ ≥ 0 and 90 applying elementaryinequalities,weobtainthat Z Ω S(v)∇(u) · ∇v = − Z Ω S(v)(u)Δv − Z Ω S′(v)(u)|∇v|2 ≤ ∥S∥ L∞(0,∞) Z Ω (u)|Δv| ≤ 1 2 Z Ω (Δv)2 + ∥S∥2 L∞(0,∞) 2 Z Ω 2(u) ≤ 1 2 Z Ω (Δv)2 + c1 ∥S∥2 L∞(0,∞) 2 Z Ω u2 ln2k+2α−2(u + e) ≤ 1 2 Z Ω (Δv)2 +  4 Z Ω u2 lnk(u + e) + c2; (5.1.41) where  is definedin(5.1.16), c2 = C() > 0 and thelastinequalitycomesfromthefactthat 2k + 2 − 2 < k and theinequality(5.1.18). Collecting(5.1.15), (5.1.41) and(5.1.19), wehave I′(t) ≤ 1 2 Z Ω (Δv)2 −  2 Z Ω u2 lnk(u + e) + c4; (5.1.42) where c4 = C() > 0. Byintegrationbypartsandelementalinequalities,itfollowsthatthereexist c5 = C() > 0 such that J′(t) := Z Ω ∇v · ∇vt = − Z Ω (Δv)2 − Z Ω |∇v|2 − Z Ω uΔv ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 + 1 2 Z Ω u2 ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 +  4 Z Ω u2 lnk(u + e) + c5: (5.1.43) The inequality(5.1.18) impliesthatthereexists c6 = C() > 0 such that Z Ω u lnk(u + e) ≤  8 Z Ω u2 lnk(u + e) + c6: (5.1.44) Collecting (5.1.40), (5.1.42), (5.1.43), and(5.1.44) weobtain y′(t) + y(t) +  8 Z Ω u2 lnk(u + e) ≤ c7 (5.1.45) for some c7 = C  r;k;; ∥S∥ L∞((0,∞))  > 0. ApplyingGronwall’sinequalitytothisleadsto y(t) ≤ max {y(0); c7}. Additionally,wealsohave:  8 Z Ω u2 lnk(u + e) ≤ c11 − y′(t): (5.1.46) 91 By integratingthepreviousinequalityfrom t to t +  and usingthefactthat y is non-negativeand bounded, wecanconcludetheproof. Now,wecanestablish Lp bounds forsolutionsinthefollowinglemma,akintoLemma 5.1.8. Lemma 5.1.11. Under theassumptioninTheorem 5.1.2, forany p > max  α 1−2α; 1 , wehavethat sup t∈(0,Tmax) Z Ω  up(·; t) + |∇v(·; t)|2p dx < ∞: (5.1.47) Proof. Wedefine (t) := 1 p Z Ω up + 1 2p Z Ω |∇v|2p; and differentiate (·) to obtain: ′(t) = Z Ω up−1  ∇ · (D(v)∇u) − ∇· (S(v)u lnα(u + e)∇v) + ru − u2 + Z Ω |∇v|2p−2∇v · ∇(Δv + u − v) := M1 +M2: (5.1.48) By integrationbyparts,wehave M1 = −2(p − 1) p Z Ω D(v)|∇u p 2 |2 + (p − 1) Z Ω S(v)up−1 lnα(u + e)∇u · ∇v + r Z Ω up −  Z Ω up+1: (5.1.49) From Lemma 5.1.10, wefindthat sup t∈(0,Tmax−τ) Z Ω u2 lnk(u + e) < ∞ for any k ∈ (1; 2 − 2 ). This,togetherwithProposition C.1.1 implies that sup t∈(0,Tmax) ∥v(·; t)∥ L∞(Ω) < ∞: (5.1.50) Therefore, itfollowsthatinf(x,t)∈Ω×(0,T ) D(v(x; t)) > 0 and −2(p − 1) p Z Ω D(v)|∇u p 2 |2 ≤ −c1 Z Ω |∇u p 2 |2; (5.1.51) 92 where c1 = 2p−2 p inf(x,t)∈Ω×(0,T ) D(v(x; t)). Since p > max  1; α 1−2α we canfix k ∈  max  2 (p + 1) p ; 1  ; 2 − 2  and Lemma 5.1.10 allows ustochoose  = min ( A1 2CGN supt∈(0,Tmax) R Ω |∇v|2 ; c1 2 supt∈(0,Tmax) R Ω u lnk(u + e) ) ; (5.1.52) where A1 is theconstantdefinedinLemma 5.1.9. ByYoung’sinequality,weobtain (p − 1) Z Ω S(v)up−1 lnα(u)∇u · ∇v = 2(p − 1) p Z Ω S(v)u p 2 lnα(u + e)∇u p 2 · ∇v ≤ c1 4 Z Ω |∇u p 2 |2 + 4(p − 1)2 p2c1 Z Ω up ln2α(u + e)|∇v|2 ≤ c1 4 Z Ω |∇u p 2 |2 +  2 Z Ω |∇v|2p+2 + c2 Z Ω up+1 ln2α(p+1) p (u + e); (5.1.53) where c2 = 8(p−1)2 p2c1ϵ . Combining(5.1.49), (5.1.51), and(5.1.53) yields M1 ≤ −3c1 4 Z Ω |∇u p 2 |2 +  2 Z Ω |∇v|2p+2 + c2 Z Ω up+1 ln2α(p+1) p (u + e) + r Z Ω up: (5.1.54) From Lemma 5.1.9, wehave M2 + A1 Z Ω |∇|∇v|p|2 + Z Ω |∇v|2p ≤ A2 Z Ω u2|∇v|2p−2 + A3 Z Ω |∇v|2p: (5.1.55) By elementaryinequalities,weobtainthat  r + 1 p Z Ω up + A2 Z Ω u2|∇v|2p−2 + A3 Z Ω |∇v|2p ≤  2 Z Ω |∇v|2p+2 + c3 Z Ω up+1 ln2α(p+1) p (u + e) + c4; (5.1.56) where c3 = C() > 0 and c4 = C() > 0. Collecting(5.1.48), (5.1.54), (5.1.55), and(5.1.56) yields ′(t) + (t) ≤ −3c1 4 Z Ω |∇u p 2 |2 − A1 Z Ω |∇|∇v|p|2 +  Z Ω |∇v|2q+2 93 + c5 Z Ω up+1 ln2α(p+1) p (u + e) + c4; (5.1.57) where c5 = c2 + c3. UsingtheGagliardo-Nirenberginterpolationinequalityfor n = 2 and thefact that supt∈(0,Tmax) R Ω |∇v(·; t)|2 dx < ∞ from Lemma 5.1.10, thereexistsapositiveconstant CGN such that:  Z Ω |∇v|2p+2 ≤ CGN Z Ω |∇|∇v|p|2 Z Ω |∇v|2 + CGN Z Ω |∇v|2 p+1 ≤ c6 Z Ω |∇|∇v|p|2 + c7; (5.1.58) where c6 = CGN supt∈(0,Tmax) R Ω |∇v(·; t)|2 and c7 = CGN supt∈(0,Tmax) 􀀀R Ω |∇v(·; t)|2 p+1. The condition 2α(p+1) p < k< 2 − 2 when p > max  α 1−2α; 1 enables ustoapplyLemma B.0.5 to obtain c5 Z Ω up+1 ln2α(p+1) p (u + e) ≤  Z Ω |∇u p 2 |2 Z Ω u lnk(u + e) +  Z Ω u p Z Ω u lnk(u + e) + c7 ≤ c8 Z Ω |∇u p 2 |2 + c9 (5.1.59) where c8 = supt∈(0,Tmax R Ω u lnk(u + e)), and c9 = c() > 0. From(5.1.57), (5.1.58), and(5.1.59), we have ′(t) + (t) ≤  c8 − 3c1 4 Z Ω |∇u p 2 |2 + (c6 − A1) Z Ω |∇|∇v|p|2 Z Ω |∇v|2 + c10; where c10 = c4 + c9. From(5.1.52), wefindthat c8 − 3c1 4 ≤ 0, and c6 − A1 ≤ 0. Itfollowsthat ′(t) + (t) ≤ c10. TheproofisfinishedbyapplyingGronwall’sinequality. ProofofTheorem 5.1.2. By usingLemma 5.1.11 for afixed p > 2, itfollowsthat sup t∈(0,Tmax) ∥u(·; t)∥ Lp(Ω) < ∞: By Lemma C.1.2, wehavethat sup t∈(0,Tmax) ∥v(·; t)∥ W1,∞(Ω) < ∞: (5.1.60) 94 Now byapplyingMoser-Alikakositerationprocedure,weobtain sup t∈(0,Tmax) ∥u(·; t)∥ L∞(Ω) < ∞: (5.1.61) This, togetherwith(5.1.60) andLemma 5.1.4 implies that Tmax = ∞, whichcompletestheproof. 5.2 Chemotaxissystemwithsuperlinearsignalproduction This sectionfocusesonstudyingblow-uppreventionofsub-logisticsourcesfor2dKeller-Segel chemotaxis systemswithsuperlinearsignalproduction.Anapplicationofaresultonparabolic gradient regularityforparabolicequationsinOrliczspacesshowsthatthepresenceofsub-logistic sources areindeedsufficientlystrongtoensuretheglobalexistenceandboundednessofsolutions. Our proofalsoreliesonseveraltechniques,includingparabolicregularityinSobolevspaces,vari- ational arguments,interpolationinequalitiesinSobolevspacesandMoseriterationmethod. 5.2.1 Introduction Weconsiderthefollowingchemotaxismodelwithsub-logisticsourcesandsuperlinearsignal production inaboundeddomainwithsmoothboundary Ω ⊂ R2: 8>>< >>: ut = ∇ · (D(v)∇u) − ∇· (uS(v)∇v) + f(u) vt = Δv − v + g(u); (5.2.1) where  ∈ {0; 1}, and 0 < D ∈ C2([0;∞)) and S ∈ C2([0;∞)) ∩W1,∞((0;∞)) such that S′ ≥ 0; (5.2.2) and thelogisticsource f(u) = ru −  u2 lnp(u + e) ; with r ∈ R; > 0; and p ∈ h 0; 1 −  2  ; (5.2.3) and thesuperlinearsignalproduction g(u) = u lnq(u + e); with q ∈ h 0; 1 −  2 − p  : (5.2.4) 95 The system(5.2.1) iscomplementedwithnonnegativeinitialconditionsinW1,∞(Ω) not identically zero: u(x; 0) = u0(x); v(x; 0) = v0(x); with x ∈ Ω; (5.2.5) and homogeneousNeumannboundaryconditionareimposedasfollows: @u @ = @v @ = 0; x ∈ @Ω; t ∈ (0; Tmax); (5.2.6) where  denotes theoutwardnormalvector. Weaimtoshowthatthepresenceofsub-logisticsourcesissufficientlystrongtoavoidblow-up solutions inasuperlinearsignalproductionchemotaxissystem.Precisely,wehavethefollowing theorems. Theorem5.2.1. Let  = 0 and thesystem (5.2.1) satisfy theassumptionsfrom (5.2.2) to (5.2.6). Thereexistsauniquepairofnonnegativefunctions (u; v) with 8>>< >>: u ∈ C0 􀀀 ¯Ω × [0;∞)  ∩ C2,1 􀀀 ¯Ω × (0;∞)  and v ∈ ∩q>2C0 ([0;∞);W1,q(Ω)) ∩ C2,1 􀀀 ¯Ω × (0;∞)  ; solving thesystem (5.2.1) in theclassicalsense.Furthermore,thissolutionisboundedinthesense that sup t>0 n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o < ∞: (5.2.7) The nexttheoremassertstheglobalexistenceandboundednessofsolutionstothefullyparabolic system (5.2.1) when  = 1. Theorem5.2.2. Let  = 1 and thesystem (5.2.1) satisfy theassumptionsfrom (5.2.2) to (5.2.6). Thereexistsauniquepairofnonnegativefunctions (u; v) with 8>>< >>: u ∈ C0 􀀀 ¯Ω × [0;∞)  ∩ C2,1 􀀀 ¯Ω × (0;∞)  and v ∈ ∩q>2C0 ([0;∞);W1,q(Ω)) ∩ C2,1 􀀀 ¯Ω × (0;∞)  ; 96 solving thesystem (5.2.1) in theclassicalsense.Furthermore,thissolutionisboundedinthesense that sup t>0 n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o < ∞: (5.2.8) The majordifficultiestoobtainauniformboundforsolutionsof(5.2.1) comefromthesuperlin- ear signalproductionofthesecondequation.Itisnotclearthatwhether ∥v(·; t)∥ L1(Ω) is uniformly bounded intime.Indeed,byintegratingthesecondequation d dt Z Ω v = Z Ω v − Z Ω u lnq(u + e); we seethatthepresenceof R Ω u lnq(u+e) has notknowntobeuniformlyboundedintime.More- over,theequi-integrabilityofthefamily R Ω u(·; t) t∈(0,Tmax) is notsufficienttopreventblow-up due tothesuperlinearsignalproduction.Inthissection,weovercomethesechallengesbyintro- ducing thefollowingfunctional: y(t) = Z Ω u(·; t) lnk(u(·; t) + e); where k > 0 will bedeterminedlater.Thisfunctional,whichhasbeenusedin[30], istheadaptation and modificationofawell-knownfunctionalcalledentropy, y(t) = Z Ω u(·; t) ln(u(·; t) + e); which hasbeenusedinvariouspaperssuchas[8, 39, 44, 71, 33, 32]. The paperisstructuredinfoursections.Section2establishesalocal-wellposednessresultfor solutions aswellasrecallsomevitalinequalities,whichwillbefrequentlyusedinsequelsections. Section 3includesaprioriestimatesforsolutionstotheparabolic-ellipticsystemwhen  = 0 and the proofofTheorem 5.2.1. InSection4,weestablishaprioriestimatesforsolutionstothefully parabolic systemwhen  = 1 and proveTheorem 5.2.2 97 5.2.2 Preliminaries The localexistenceanduniquenessofnon-negativeclassicalsolutionstothesystem(5.2.1) can be establishedbyadaptingandadjustingthefixedpointargumentandstandardparabolicregularity theory.Forfurtherdetails,wereferthereaderto[21, 29, 58]. Forconvenience,weadoptLemma 4.1 from[61]. Lemma 5.2.3. Let Ω ⊂ Rn, where n ≥ 2 be aboundeddomainwithsmoothboundary,andthe system (5.2.1) satisfy theconditionsfrom (5.2.2) to (5.2.6). Thenthereexist Tmax ∈ (0;∞] and functions 8>>< >>: u ∈ C0 􀀀 ¯Ω × [0; Tmax)  ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  and v ∈ T q>2 C0 ([0; Tmax);W1,q(Ω)) ∩ C2,1 􀀀 ¯Ω × (0; Tmax)  (5.2.9) such that u > 0 and v > 0 in ¯Ω ×(0;∞), that (u; v) solves (5.2.1) classically in Ω×(0; Tmax), and that if Tmax < ∞, then lim sup t→Tmax n ∥u(·; t)∥ L∞(Ω) + ∥v(·; t)∥ W1,∞(Ω) o = ∞: (5.2.10) The followinglemmaisessentialtoobtainan L2 ln(L + e) estimate forthesolutionof(5.2.1) when  = 1. Lemma 5.2.4. Let Ω ⊂ R2 be aboundeddomainwithsmoothboundary,and k > 1 2 . Thenthere exists C > 0 such thatforeach  > 0, onecanpick C() > 0 such that Z Ω u3 ln3/2(u + e) ≤  Z Ω |∇u|2 ln(u + e) Z Ω u lnk(u + e) +  Z Ω u lnk(u + e) + c Z Ω u ln1/2(u + e) 3 : (5.2.11) holds for all u ∈ C2(Ω¯). Proof. WeapplyLemma B.0.9 with G(s) := s lnk−1/2(s + e), todeducethatforany  > 0, there exists c = c() > 0 such that Z Ω  u ln1/2(u + e) 3 ≤  Z Ω |∇(u ln1/2(u + e))|2 Z Ω u lnk(u + e) 98 + c Z Ω u ln1/2(u + e) 3 : (5.2.12) Notice that Z Ω |∇(u ln1/2(u + e))|2 ≤ Z Ω 2|∇u|2 ln(u + e) + u (u + e) ln1/2(u + e) |∇u|2 ≤ c Z Ω |∇u|2 ln(u + e); (5.2.13) Where thelastinequalitycomesfrom Z Ω u (u + e) ln1/2(u + e) |∇u|2 ≤ Z Ω |∇u|2 ln(u + e) Finally,wemakeuseof(5.2.12) and(5.2.13) tocompletetheproof. 5.2.3 Parabolic-elliptic In thissection,weassumethat (u; v) is asolutionofthesystem(5.2.1) with  = 0, underthe conditions from(5.2.2) to(5.2.6). Letusbeginwiththefollowingprioriestimate,whichwillbe used toobtain Lm bounds forthesolution. Lemma 5.2.5. For any k ≥ 1 , then sup t∈(0,Tmax) Z Ω u lnk (u + e) < ∞: (5.2.14) Proof. Wemakeuseofthefirstequationandintegrationbypartstoobtain d dt Z Ω u lnk(u + e) = Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  (∇ · (D(v)∇u − uS(v)∇v) + f(u)) = −k Z Ω D(v) lnk−1(u + e) u + e |∇u|2 − k(k − 1) Z Ω D(v)u lnk−2(u + e) (u + e)2 |∇u|2 − k Z Ω eD(v) lnk−1(u + e) (u + e)2 |∇u|2 + Z Ω S(v)∇(u) · ∇v + Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  f(u) ≤ Z Ω S(v)∇(u) · ∇v + Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  f(u) (5.2.15) 99 where (l) = Z l 0  k s lnk−1(s + e) s + e + k(k − 1) s2 lnk−2(s + e) (s + e)2 + k eu lnk−1(s + e) (s + e)2  ds ≤ c1l lnk−1(l + e); (5.2.16) with c1 := k2 + k. This,togetherwithintegrationbyparts,thecondition S′ ≥ 0 and elementary inequalities, followsthat Z Ω S(v)∇(u) · ∇v = − Z Ω S(v)(u)Δv − Z Ω S′(v)(u)|∇v|2 ≤ ∥S∥ L∞([0,∞)) Z Ω (u)u lnq(u + e) ≤ c2 Z Ω u2 lnk+q−1(u + e) ≤  2 Z Ω u2 lnk−p(u + e) + c3; (5.2.17) where c2 = c1 ∥S∥ L∞([0,∞)), c3 = C(; k; ∥S∥ L∞([0,∞))) > 0 and thelastinequalitycomesfrom the factthat p < 1 − q and forany  > 0, thereexist c() > 0 such that ua1 lnb1(u + e) ≤ ua2 lnb2(u + e) + c(); where a1; a2; b1; b2 are realnumberssuchthat a1 < a2. Thisalsoimpliesthatthereexistsapositive constant c4 depending on  such that  lnk (u + e) + k lnk−1 (u + e) u + e  f(u) + u lnk(u + e) ≤ (r + 1)u lnk(u + e) + rk lnk−1(u + e) − u2 lnk−p(u + e) ≤  4 Z Ω u2 lnk−p(u + e) + c4: (5.2.18) Integrating (5.2.18) bothsidesentailsthat Z Ω  lnk (u + e) + k lnk−1 (u + e) u + e  f(u) + Z Ω u lnk(u + e) ≤  4 Z Ω u2 lnk−p(u + e) + c5; (5.2.19) where c5 = c4|Ω|. Collecting(5.2.15), (5.2.17) and(5.2.19) yields d dt Z Ω u lnk(u + e) + Z Ω u lnk(u + e) ≤ c6; (5.2.20) where c6 = c3 + c5. This,togetherwithGronwall’sinequalitycompletestheproof. 100 The nextlemmaprovides Lm bounds forthesolutionof(5.2.1) forany m ≥ 1. Lemma 5.2.6. For any m > 1 , then sup t∈(0,Tmax) Z Ω um < ∞: (5.2.21) Proof. Multiplying thefirstequationof(5.2.1) by um−1 and usingintegrationbypartsyields 1 m d dt Z Ω um = Z Ω um−1 (∇ · (D(v)∇u − uS(v)∇v) + f(u)) = −4(m − 1) m2 Z Ω D(v)|∇u m 2 |2 + m − 1 m Z Ω S(v)∇um · ∇v + Z Ω um−1f(u) (5.2.22) Let f = u lnq(u + e), weobtain Z Ω f ln(f + e) = Z Ω u lnq(u + e) ln (u lnq(u + e) + e) ≤ Z Ω u lnq(u + e) ln ((u + e) lnq(u + e)) ≤ (q + 1) Z Ω u lnk(u + e); (5.2.23) for any k >q + 1. NowbyapplyingLemma 5.2.5 with afixed k >q + 1, andLemma 5.1.6 , we find that v is uniformlyboundedintime.Therefore,degeneracyofthediffusiontermsiseliminated since inf(x,t)∈Ω×(0,Tmax) D(v(x; t)) := c1 > 0. Thus,wehave −4(m − 1) m2 Z Ω D(v)|∇u m 2 |2 ≤ −c2 Z Ω |∇u m 2 |2; (5.2.24) where c2 = 4(m−1)c1 m2 . Usingintegrationbyparts,thesecondequationof(5.2.1) andnonnegativity of S′ and v deduces that m − 1 m Z Ω S(v)∇um · ∇v = −m − 1 m Z Ω S(v)umΔv − m − 1 m Z Ω S′(v)um|∇v|2 ≤ −m − 1 m Z Ω S(v)um (u lnq(u + e) − v) ≤ c3 Z Ω um+1 lnq(u + e); (5.2.25) 101 where c3 = −(m−1)∥S∥ L ∞ ([0,∞)) m . ApplyingLemma 5.2.5 with k = 1 and Lemma B.0.5 with  = − c2 2c3 supt∈(0,Tmax) R Ω u ln(u + e) ; yields c3 Z Ω um+1 lnq(u + e) ≤ c3 Z Ω |∇u m 2 |2 · Z Ω u ln(u + e) +  Z Ω u m · Z Ω u ln(u + e) + c4 ≤ c2 2 Z Ω |∇u m 2 |2 + c5; (5.2.26) where c4 = C() > 0 and c5 = c4+supt∈(0,Tmax) R Ω u ln(u+e)·supt∈(0,Tmax) 􀀀R Ω u m. Byelementary inequalities, thereexistsapositiveconstant c6 = C(r;m;p;) such that 1 m Z Ω um + Z Ω um−1f(u) ≤ − 2 Z Ω um+1 lnp(u + e) + c6: (5.2.27) Collecting (5.2.22), (5.2.25), (5.2.26), and(5.2.27) entailsthat 1 m d dt Z Ω um + 1 m Z Ω um ≤ c7; where c7 = c5 + c6. This,togetherwithGronwall’sinequalityproves(5.2.21), whichfinishesthe proof. WearenowreadytoproveTheorem 5.2.1. ProofofTheorem 5.2.1. It followsfromLemma 5.2.6 that u ∈ L∞ ((0; Tmax); Lm(Ω)) for anym > 1. Bystandardellipticregularitytheory,weobtainthat v ∈ L∞ ((0; Tmax);W1,∞(Ω)). Byapplying Moser-Alikakositeration(seee.g[53, 2, 1] ),itfollowsthat u ∈ L∞ ((0; Tmax); L∞(Ω)). Now applying theextensibityofsolutions(5.2.10) yieldsthat Tmax = ∞. Therefore,(5.2.7) isproved, which finishestheproof. 5.2.4 Fullyparabolic In thissection,weconsiderasolution (u; v) of thesystem(5.2.1) with  = 1, underthecondi- tions from(5.2.2) to(5.2.6). Letuscommencewiththefollowingprioriestimate,similartoLemma 5.2.5. 102 Lemma 5.2.7. If p < 1 < k< 2 − p, and 2q + p < 1 then sup t∈(0,Tmax) Z Ω 􀀀 u lnk (u + e) + |∇v|2 + sup t∈(0,Tmax−τ) Z t+τ t Z Ω u2 lnk−p(u + e) < ∞; (5.2.28) where  = min  1; Tmax 2 . Proof. Wedefine y(t) := Z Ω u lnk (u + e) + 1 2 Z Ω |∇v|2; and differentiate y(·) to obtain y′(t) = Z Ω  lnk (u + e) + ku lnk−1 (u + e) u + e  ut + ∇v · ∇vt (5.2.29) By integrationbyparts,thefirsttermof(5.2.29) isexpressedasin(5.2.15). Wehave Z Ω S(v)∇(u) · ∇v = − Z Ω S(v)(u)Δv − Z Ω S′(v)(u)|∇v|2 ≤ ∥S∥ L∞([0,∞)) Z Ω (u)|Δv|; where thelastinequalitycomesfromtheassumption S′ ≥ 0 and  is definedin(5.2.16). Recalling the upperboundfor  as in(5.2.16), wehave (l) ≤ c1l lnk−1(l + e): This, togetherwithYoung’sinequalitytothisyields Z Ω S(v)∇(u) · ∇v ≤ 1 2 Z Ω (Δv)2 + ∥S∥ L∞([0,∞)) 2 Z Ω 2(u) ≤ 1 2 Z Ω (Δv)2 + c2 Z Ω u2 ln2k−2(u + e) ≤ 1 2 Z Ω (Δv)2 +  4 Z Ω u2 lnk−p(u + e) + c3; (5.2.30) where c2 = c1∥S∥ L ∞ ([0,∞)) 2 and c3 = C(:k;p) > 0. Collecting(5.2.15),(5.2.17),(5.2.19) and (5.2.30) impliesthat d dt Z Ω u lnk(u + e) + Z Ω u lnk(u + e) + 3 4 Z Ω u2 lnk−p(u + e) ≤ 1 2 Z Ω (Δv)2 + c4; (5.2.31) 103 where c4 = C(; r;k;p; ∥S∥ L∞([0,∞))) > 0. Byintegrationbypartsandelementaryinequalities, the secondtermof(5.2.29) canbehandledasfollows: Z Ω ∇v · ∇vt = − Z Ω (Δv)2 − Z Ω |∇v|2 − Z Ω u lnq(u + e)Δv ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 + 1 2 Z Ω u2 ln2q(u + e) ≤ −1 2 Z Ω (Δv)2 − Z Ω |∇v|2 +  2 Z Ω u2 lnk−p(u + e) + c5; 2q 0. From(5.2.31) and(5.2.32), weobtainthat y′(t) + y(t) +  2 Z Ω u2 lnk−p(u + e) ≤ c6; (5.2.33) where c6 = c4 + c5. ByapplyingGronwall’sinequality,wededucethat y(t) ≤ max {y(0); c6}. Additionally,wehave  2 Z Ω u2 lnk−p(u + e) ≤ c6 − y′(t): (5.2.34) By integratingthepreviousinequalityfrom t to t +  and usingthefactthat y is bounded,wecan conclude theproof. Thanks toLemma 5.2.7 and C.1.1, wecannowestablishan L∞ bound for v, whichissubse- quently usedtoeliminatethedegeneracyofthediffusionterm. Lemma 5.2.8. If 1 + p + 2q 1 will bedeterminedlater. Notice that u lnq(u + e) ≤ (u + e)2, wehave L(f) = lnλ(u lnq(u + e)) ≤ 2λ lnλ(u + e): Now,wewant sup t∈(0,T−τ) Z t+τ t Z Ω f2L(f) < ∞; (5.2.35) 104 which isindeedtruewhen 2q +  ≤ k − p, since Z t+τ t Z Ω f2L(f) ≤ c Z t+τ t Z Ω u2 ln2q+λ(u + e) ≤ c Z t+τ t Z Ω u2 lnk−p(u + e): (5.2.36) Now wefixwhere  := k − p − 2q > 1 and completetheproofbyapplyingProposition C.1.1 . In contrasttotheparabolic-ellipticscenariowhere  = 0, thedirectderivationof Lm bounds for u from theaprioriestimateinLemma 5.2.7 is notfeasible.Instead,werelyontheassistance of thesubsequentlemma,whichfunctionsasanintermediateestimatefacilitatingtheconnection between thetwoequationspresentedin(5.2.1). Lemma 5.2.9. Thereexistpositiveconstants A1;A2;A3 such that 1 4 d dt Z Ω |∇v|4 + A1 Z Ω ∇|∇v|2 2 + Z Ω |∇v|4 ≤ A2 Z Ω u2 ln2q(u + e)|∇v|2 + A3 Z Ω |∇v|4 (5.2.37) Proof. Wemakeuseofthefollowingpoint-wiseidentity ∇v · ∇Δv = 1 2 Δ(|∇v|2) − |D2v|2 to obtain 1 4 d dt Z Ω |∇v|4 + Z Ω |∇v|4 = −c1 Z Ω |∇|∇v|2|2 − Z Ω |∇v|2|D2v|2 + Z Ω |∇v|2∇v · ∇(u lnq(u + e)) + c2 Z ∂Ω @|∇v|2 @ |∇v|2; (5.2.38) where c1; c2 are positiveconstants.Theinequality ∂|∇v|2 ∂ν ≤ M|∇v|2; (see [41][Lemma 4.2])for some M > 0 depending onlyon Ω, impliesthat c2 Z ∂Ω @|∇v|2 @ |∇v|2 dS ≤ c2M Z ∂Ω |∇v|4 dS: Let g := |∇v|2 and applytraceembeddingtheorem W1,1(Ω) −→ L1(@Ω) together withYoung’s inequality,thereexistpositiveconstants C and c3 such that c2M Z ∂Ω g2 dS ≤ C Z Ω g|∇g| + C Z Ω g2 105 ≤ c1 2 Z Ω |∇g|2 + c3 Z Ω g2; (5.2.39) Therefore, wehave c2M Z ∂Ω |∇v|4 dS ≤ c1 2 Z Ω |∇|∇v|2|2 + c3 Z Ω |∇v|4: (5.2.40) Applying thepointwiseinequality (Δv)2 ≤ 2|D2v|2 to (5.2.38) yields 1 4 d dt Z Ω |∇v|4 + Z Ω |∇v|4 ≤ −c1 2 Z Ω |∇|∇v|2|2 − 1 2 Z Ω |∇v|2|Δv|2 + Z Ω |∇v|2∇v · ∇(u lnq(u + e)) + c3 Z Ω |∇v|4 (5.2.41) By integrationbypartsandelementalinequalities,thereexistconstants c4 > 0 and c5 > 0 in such a waythat Z Ω |∇v|2∇v · ∇(u lnq(u + e)) = − Z Ω u lnq(u + e)|∇v|2Δv − c4 Z Ω u lnq(u + e)∇|∇v|2 · ∇v ≤ 1 2 Z Ω (Δv)2|∇v|2 + c1 4 Z Ω |∇|∇v|2|2 + c5 Z Ω u2 ln2q(u + e)|∇v|2: (5.2.42) Combining (5.2.41) and(5.2.42) yields(5.2.37). One mayemployLemma B.0.5 to establishan Lm bound for u, where m ∈ (2; 3 − 2q). How- ever,inthiscontext,analternativemethodologyisadopted,relyingonLemma 5.2.4. Thisapproach involves initiallyderivingan L2 ln(L + e) bound andsubsequentlyutilizingittoestablishan L4 bound for u. Lemma 5.2.10. If k >p + 2q + 1 such that sup t∈(0,Tmax) Z Ω u lnk(u + e) < ∞; then wehave sup t∈(0,Tmax) Z Ω 􀀀 u2 ln(u + e) + |∇v|4 < ∞: 106 Proof. Wedenote y(t) := Z Ω u2 ln(u + e) + 1 4 Z Ω |∇v|4; and differentiate y to obtain y′(t) = Z Ω  2u ln(u + e) + u2 u + e  ut + Z Ω |∇v|2∇v · ∇vt := I + J: (5.2.43) Weusethefirstequationof(5.2.1), andintegrationbypartstoestimate I I := Z Ω  2u ln(u + e) + u2 u + e  (∇ · (D(v)∇u − uS(v)∇v + f(u))) = − Z Ω D(v)  2 ln(u + e) + 2u u + e + u2 + 2ue (u + e)2  |∇u|2 + Z Ω uS(v)  2 ln(u + e) + 2u u + e + u2 + 2ue (u + e)2  ∇u · ∇v + Z Ω  2u ln(u + e) + u2 u + e  f(u) := I1 + I2 + I3: (5.2.44) Lemma 5.2.8 implies that v is boundedatalltime,whichentailsthatinf(x,t)∈Ω×(0,T ) D(v(x; t)) := > 0. Therefore, I1 is boundedby I1 ≤ − Z Ω ln(u + e)|∇u|2: (5.2.45) Now, I2 can becontrolledbyusingelementaryinequalities I2 := Z Ω uS(v)  2 ln(u + e) + 2u u + e + u2 + 2ue (u + e)2  ∇u · ∇v ≤ c1 Z Ω u ln(u + e)|∇u||∇v| ≤ 2 Z Ω |∇u|2 ln(u + e) + c2 Z Ω u2 ln(u + e)|∇v|2 (5.2.46) where c1 = 5 ∥S∥ L∞([0,∞)) and c2 = c21 4α. UsingYoung’sinequalitywith  > 0 yields c2 Z Ω u2 ln(u + e)|∇v|2 ≤  Z Ω |∇v|6 + c3 Z Ω u3 ln3 2 (u + e); (5.2.47) where c3 = C() > 0. Byusingelementaryinequalities,onecanfindapositiveconstant c4 = C(; r;p) such that I3 + Z Ω u2 ln(u + e) = Z Ω  2u ln(u + e) + u2 u + e  ru − u2 lnp(u + e)  + Z Ω u2 ln(u + e) 107 ≤ c4: (5.2.48) Collecting from(5.2.45) to(5.2.48) yields I + Z Ω u2 ln(u + e) + 2 Z Ω |∇u|2 ln(u + e) ≤  Z Ω |∇v|6 + c3 Z Ω u3 ln3/2(u + e) + c4: (5.2.49) From Lemma 5.2.9 and applyingYoung’sinequality,wehave 1 4 d dt Z Ω |∇v|4 + A1 Z Ω ∇|∇v|2 2 + Z Ω |∇v|4 ≤ A2 Z Ω u2 ln2q(u + e)|∇v|2 + A3 Z Ω |∇v|4 ≤  Z Ω |∇v|6 + c5 Z Ω u3 ln3q(u + e) + c6 ≤  Z Ω |∇v|6 + c5 Z Ω u3 ln3/2(u + e) + c6; (5.2.50) where c5 = C() > 0, c6 = C() > 0, andthelastinequalitycomesfromthefactthat 2q < 1. Collecting (5.2.49) and(5.2.50) yields y′(t) + y(t) + A1 Z Ω ∇|∇v|2 2 + 2 Z Ω |∇u|2 ln(u + e) ≤ 2 Z Ω |∇v|6 + c7 Z Ω u3 ln3/2(u + e) + c8 (5.2.51) where c7 = c3 + c5 and c8 = c4 + c6. FromLemma B.0.2, thereexistsapositiveconstant c9 such that Z Ω |∇v|6 ≤ c9 Z Ω |∇|∇v|2|2 · Z Ω |∇v|2 + c9 Z Ω |∇v|2 3 : (5.2.52) Lemma 5.2.7 asserts that R Ω |∇v|2 is uniformlyboundedintime,thereforefrom(5.2.52) weobtain 2 Z Ω |∇v|6 ≤ c10 Z Ω |∇|∇v|2|2 + c11; (5.2.53) where c10 = 2c9 supt∈(0,Tmax) R Ω |∇v|2 and c11 = c9  supt∈(0,Tmax) R Ω |∇v|2 3 . Nowwefix  = A1 2c10 and applyLemma 5.2.4 with  = 4c7 supt∈(0,Tmax) R Ω u lnk(u + e) 108 to deducethat c7 Z Ω u3 ln3/2(u + e) ≤ c7 Z Ω |∇u|2 ln(u + e) · Z Ω u lnk(u + e) + c7 Z Ω u lnk(u + e) + c12 Z Ω u ln1/2(u + e) 3 ≤ 4 Z Ω |∇u|2 ln(u + e) + c13; (5.2.54) where c12 = C() > 0 and c13 = c7 sup t∈(0,Tmax) Z Ω u lnk(u + e) + c12 sup t∈(0,Tmax) Z Ω u ln1/2(u + e) !3 : Collecting (5.2.51), (5.2.53), and(5.2.54) yields y′(t) + y(t) ≤ c14; (5.2.55) where c14 = c8 + c13. This,togetherwithGronwall’sinequalitycompletestheproof. Wearereadtoderivean L4 bound for u by employingastandardtestingprocedure. Lemma 5.2.11. If (u; v) is asolutionofthesystem (5.2.1) such that sup t∈(0,T ) Z Ω u2 ln(u + e) < ∞; then sup t∈(0,T ) Z Ω u4 < ∞: Proof. Multiplying thefirstequationof(5.2.1) to u3 and usingintegrationbyparts,wehave d dt Z Ω u4 4 = −3 Z Ω D(v)u2|∇u|2 + 3 Z Ω S(v)u3∇u · ∇v + Z Ω u3f(u): (5.2.56) From Lemma 5.2.8 entails that v is boundedatalltime,wehaveinf(x,t)∈Ω×(0,T ) D(v(x; t)) := c1 > 0. Therefore,weobtain −3 Z Ω D(v)u2|∇u|2 ≤ −3c1 Z Ω u2|∇u|2: (5.2.57) 109 By usingHolder’sinequality,wehave 3c1 Z Ω S(v)u3∇u · ∇v ≤ c1 Z Ω u2|∇u|2 + 9c1 4 Z Ω u4|∇v|2: (5.2.58) Now wefindthat g(u) ∈ L∞ ((0; Tmax); L2(Ω)) since Z Ω g2(u) = Z Ω u2 ln2q(u + e) ≤ Z Ω u2 ln(u + e) ≤ C for all t ∈ (0; Tmax). Therefore,Lemma C.1.2 implies that v ∈ L∞ 􀀀 (0; Tmax);W1,λ(Ω)  for any  ∈ [1;∞), whichmeansthatsupt∈(0,T ) R Ω |∇v|λ < ∞. Now,weestimatethelasttermoftheright hand sideof(5.2.58) byusingHolder’sinequalityandthenYoung’sinequalityasfollows 9c1 4 Z Ω u4|∇v|2 ≤ 9c1 4 Z Ω u9/2 8/9 Z Ω |∇v|18 1/9 ≤ c2 Z Ω u9/2 8/9 ≤ c2 Z Ω u9/2 + c3 ≤  2 Z Ω u5 lnp(u + e) + c4: (5.2.59) where c2 = 9c1 4 supt∈(0,T ) R Ω |∇v|18, c3 > 0, and c4 = C(; p) > 0 By applyingYoung’sinequality again, onecanverifythat Z Ω u3f(u) + 1 4 Z Ω u4 ≤ − 2 Z Ω u5 lnp(u + e) + c5; (5.2.60) where c5 = C(; p) > 0. Collecting(5.2.59) and(5.2.60) yields 1 4 d dt Z Ω u4 + 1 4 Z Ω u4 ≤ c6; where c6 = c4 + c5. TheproofisfinishedbyapplyingGronwall’sinequality. Now weareinthepositiontoproveTheorem 5.2.2. ProofofTheorem 5.2.2. First, fromtheassumption 2q + p 0. Now,weuseLemma 5.2.10 and Lemma 5.2.11 to obtain sup t∈(0,T ) Z Ω u4 < ∞: This, togetherwith R Ω g3(u) ≤ R Ω u4, entailsthat g(u) ∈ L∞ ((0; Tmax); L3(Ω)). Therefore,by applying Lemma C.1.2, wededucethat v ∈ L∞ ((0; Tmax);W1,∞(Ω)). 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Zhengetal.“Anewresultforglobalexistenceandboundednessofsolutionstoaparabolic– parabolic Keller–Segelsystemwithlogisticsource”.In: Journal ofMathematicalAnalysis and Applications 462 (June2018),pp.1–25. 117 APPENDIX A NOTATIONS Weintroducesomenotationsusedthroughoutthisbook. Ω is anopenboundedsetin Rn with smoothboundary. ΩT := Ω × (0; T).  is theoutwardnormalvector. WefollowsomedefinitionsofHoldercontinuousspacesgivenin[37]. For k ∈ N, and ∈ (0; 1], we definethefollowingnormsandseminorms: [f]k+γ,ΩT := X |β|+2j=k sup (x,t)̸=(y,s)∈ΩT |Dβ xDj t (f(x; t) − f(y;s)) | ∥x − y∥γ + |t − s| γ 2 ; ⟨f⟩ k+γ,ΩT := X |β|+2j=k−1 sup (x,t)̸=(x,s)∈ΩT |Dβ xDj t (f(x; t) − f(x; s)) | |t − s| 1+γ 2 ; |f|k+γ,ΩT := [f]k+γ,ΩT + ⟨f⟩ k+γ,ΩT ; ∥f∥ Ck(ΩT ) := X |β|+2j≤k sup (x,t)∈ΩT |Dβ xDj t f(x; t)| ∥f∥ Ck+γ(ΩT ) := ∥f∥ Ck(ΩT ) + |f|k+γ,ΩT Now wedefinethefollowingfunctionalspaces Ck(ΩT ) :=  f : Dβ xDj t f is continuousin ΩT for | | + 2j = k Ck+γ(ΩT ) := n f ∈ Ck(ΩT ) : ∥f∥ Ck+γ(ΩT ) < ∞ o : One canverifythat Ck(¯ΩT ) and Ck+γ(¯ΩT ) are Banachspaces.Thesmoothnessconditionofbound- ary isnecessarytoguaranteetheinclusion Ca(¯ΩT ) ⊂ Cb(¯ΩT ) for b >a ≥ 0, sinceitisnottrue in generaldomain Ω. Moreover, Ca(¯ΩT ) is compactlyembeddingin Cb(¯ΩT ) for any a ≥ 1 and a >b ≥ 0 (see [15][Lemma 6.36]). Lemma A.0.1. Assume that ∈ (0; 1], thefunctions f(x) = |x|γ and g(x) = |x|1+γ belong to Cγ(R) and C1+γ(R) respectively. 118 Proof. Wehave |x| ≤|y| + |x − y| ≤ (|y|γ + |x − y|γ) 1 γ ; (A.0.1) where thelastinequalitycomesfrom as + bs ≤ (a + b)s; s ≥ 1; a;b ≥ 0: From (A.0.1), wehave |f(x) − f(y)| = ||x|γ − |y|γ| ≤|x − y|γ; (A.0.2) which impliesthat f is in Cγ(R). Nowweshowthat g is in C1+γ(R). Differentiating g, weobtain g′(x) = 8>>>>>>< >>>>>>: (1 + )|x|γ; x> 0; 0; x = 0; −(1 + )|x|γ; x< 0: (A.0.3) When xy = 0, onecanverifythat |g′(x) − g′(y)| ≤ (1 + )|x − y|γ: (A.0.4) When xy > 0, wemakeuseofthefactthat f ∈ Cγ(R) to have |g′(x) − g′(y)| ≤|x − y|γ: (A.0.5) When xy < 0, wehave |g′(x) − g′(y)| = (1+ )(|x|γ + |y|γ) ≤ 2(1 + )|x − y|γ: (A.0.6) From (A.0.4), (A.0.5), and(A.0.6), weconcludethat g ∈ C1+γ(R). Lemma A.0.2. If u ∈ C1+γ(¯ΩT ), and f ∈ C1+γ(R) with ∈ (0; 1], then f(u) ∈ C1+γ(¯ΩT ). Proof. It isstraightforwardtoverifythat f(u) is in C1(¯ΩT ). For | | = 1, wehave |Dβ x (f(u(x; t)) − f(u(y;s))) | ≤|Dβ x (f(u(x; t)) − f(u(x; s))) | 119 +|Dβ x (f(u(x; s)) − f(u(y;s))) |: (A.0.7) Since f ∈ C1+γ(R) and u ∈ C1+γ(¯ΩT ), weobtain |Dβ x (f(u(x; t)) − f(u(x; s))) | = |f′(u(x; t))Dβ xu(x; t) − f′(u(x; s))Dβ xu(x; s)| ≤ |f′(u(x; t))||Dβ x (u(x; t) − u(x; s))| + |f′(u(x; t)) − f′(u(x; s))||Dβ xu(x; s)| ≤ C1|t − s|γ/2 + C2|u(x; t) − u(x; s)|γ ≤ C1|t − s|γ/2 + C2|t − s|γ ≤ (C1 + C2Tγ/2)|t − s|γ/2: (A.0.8) Similarly,weobtain |Dβ x (f(u(x; s)) − f(u(y;s))) | ≤ C ∥x − y∥γ : (A.0.9) From (A.0.7), (A.0.8), and(A.0.9), wehave sup (x,t)̸=(y,s)∈¯ΩT |Dβ x (f(u(x; t)) − f(u(y;s))) | ∥x − y∥γ + |t − s|γ/2 < ∞: (A.0.10) By similararguments,wealsohave sup (x,t)̸=(y,s)∈¯ΩT |Dβ x (f(u(x; t)) − f(u(x; s))) | |t − s|(1+γ)/2 < ∞: (A.0.11) Finally,(A.0.10) and(A.0.11) imply f(u) ∈ C1+γ(¯ΩT ). For any p ∈ [1;∞), wedefine Lp spaces Lp(Ω) := ( f is measurablein Ω : ∥f∥ Lp(Ω) := Z Ω |f|p dx 1 p < ∞ ) ; and L∞(Ω) := n f is measurablein Ω : ∥f∥ L∞(Ω) := ess-supx∈Ω |f(x)| < ∞ o : WedefineSobolevspacesforany 1 ≤ p ≤ ∞ Wk,p(Ω) := n f ∈ Lp(Ω) : ∥Dαf∥ Lp(Ω) < ∞ o : where is amulti-indexsuchthat | | ≤ k, and Dαf is aweakderivativeof f. 120 APPENDIX B INEQUALITIES Wecollectsomeusefulinequalitiesfrequentlyusedthroughthisthesis.LetusbeginwithYoung’s inequality: Lemma B.0.1 (Young’sinequality). For any  > 0, p > 1, and a; b> 0, thefollowinginequality holds ab ≤ as + s − 1 s (s) 1 1−s b s s−1 : (B.0.1) Next, letusintroduceanextendedversionoftheGagliardo-Nirenberginterpolationinequality, which wasestablishedin[36][Lemma 2.3]. Lemma B.0.2 (Gagliardo-Nirenberginterpolationinequality). Let Ω be aboundedandsmooth domain of Rn with n ≥ 1. Let r ≥ 1, 0 < q ≤ p < ∞, s ≥ 1. Thenthereexistsaconstant CGN > 0 such that ∥f∥p Lp(Ω) ≤ CGN  ∥∇f∥pa Lr(Ω) ∥f∥p(1−a) Lq(Ω) + ∥f∥p Ls(Ω)  for all f ∈ Lq(Ω) with ∇f ∈ (Lr(Ω))n, and a = 1 q −1 p 1 q+ 1 n −1 r ∈ [0; 1]. Consequently,thenextlemmaisderivedasfollows: Lemma B.0.3. If Ω be aboundedandsmoothdomainof Rn with n ≥ 1, thenthereexistsapositive constant C depending onlyon Ω such thatforany f ∈ W1,2(Ω) the followinginequality Z Ω f2 ≤ C Z Ω |∇f|2 + C  n 2 Z Ω |f| 2 (B.0.2) holds forall  ∈ (0; 1). Proof. The LemmafollowsfromLemma B.0.2 by choosing p = r = 2 and q = s = 1 and Young’s inequality. 121 The nextlemmaprovidesanessentialinequalityusedtoabsorbnonlinearchemo-attractants term intothediffusionterm.Itisadirectconsequenceof[66][Corollary 1.2],howeverforthe convenience, weprovidethedetailproofhere. Lemma B.0.4. If Ω ⊂ R2 is aboundeddomainwithsmoothboundary,thenforeach m > 0 and ≥ 0 thereexists C = C(m; ) > 0 with thepropertythatwhenever  ∈ C1(¯Ω) is positivein ¯Ω Z Ω m+1 lnγ( + e) ≤ C Z Ω  lnγ( + e) Z Ω |∇ m 2 |2  + C Z Ω  m Z Ω  lnγ( + e)  : (B.0.3) Proof. By applyingSobolev’sinequalitywhen n = 2, thereexistsapostiveconstant c1 such that Z Ω m+1 lnγ( + e) ≤ c1 Z Ω ∇   m+1 2 lnγ 2 ( + e)  2 + c1 Z Ω  ln γ m+1 ( + e) m+1 (B.0.4) By usingelementaryinequalities,onecanverifythat ∇   m+1 2 lnγ 2 ( + e)  ≤ c2 1 2 lnγ 2 ( + e)|∇ m 2 |; where c2 = C(m; ) > 0. This,togetherwithHolder’sinequalityleadsto c1 Z Ω ∇   m+1 2 lnγ 2 ( + e)  2 ≤ c3 Z Ω |∇ m 2 |2 · Z Ω  lnγ( + e); (B.0.5) where c3 = c1c2. ByHolder’sinequality,wededucethat c1 Z Ω  ln γ m+1 ( + e) m+1 ≤ c1 Z Ω  m Z Ω  lnγ( + e)  : (B.0.6) Collecting (B.0.4), (B.0.5) and(B.0.6) implies(B.0.3), whichfinishestheproof. As aconsequence,wehavethefollowinginterpolationinequalitywitharbitry  parameters. Lemma B.0.5. Assume that Ω ⊂ R2 is aboundeddomainwithsmoothboundaryand p > 0, > ≥ 0. Foreach  > 0, thereexists C = C(; ; ) > 0 such that Z Ω m+1 lnξ( + e) ≤  Z Ω  lnγ( + e) Z Ω |∇ m 2 |2  +  Z Ω  m Z Ω  lnγ( + e)  + C: (B.0.7) 122 Proof. Since > ≥ 0, onecanverifythatforany  > 0, thereexists c1 = c(;; ) > 0 such that forany a ≥ 0 we have am+1 lnξ(a + e) ≤ am+1 lnγ(a + e) + c1: (B.0.8) This entailsthat Z Ω m+1 lnξ( + e) ≤  Z Ω m+1 lnγ( + e) + c1|Ω|: (B.0.9) Now foranyfixed , wechoose  = ϵ C where C as inLemma B.0.4, andapply(B.0.3) tohavethe desire inequality(B.0.7). The followinglemmaprovidesestimatesontheboundary(seeLemma5.3in[40]): Lemma B.0.6. Assume that Ω is aconvexboundeddomain,andthat f ∈ C2(¯Ω) satisfies ∂f ∂ν = 0 on @Ω. Then @|∇f|2 @ ≤ 0 on @Ω: The nextlemmaprovidesestimatesontheboundaryofnonconvexboundeddomain (see [41][Lemma 4.2]). Lemma B.0.7. Assume that Ω is boundedandlet w ∈ C2(¯Ω) satisfy ∂w ∂ν = 0 on @Ω. Thenwehave @|∇w|2 @ ≤ 2|∇w|2; (B.0.10) where  = (Ω) > 0 is anupperboundforthecurvaturesof @Ω. Next letusderiveanestimateforaparticularboundaryintegralthatenablesustocoverpossibly non-convex domains. Lemma B.0.8. Let Ω be aboundeddomainwithsmoothboundary,let q ∈ [1; ∞). Thenforany  > 0 thereis Cθ > 0 such thatforany f ∈ C2(¯Ω) satisfying ∂f ∂ν = 0 on @Ω and ,theinequality Z ∂Ω |∇f|2(q−1)∇(|∇f|2) ·  ≤  Z Ω |∇(|∇f|q)|2 + c() Z Ω |∇f|2q: holds. 123 Proof. From Lemma B.0.7, itfollowsthat Z ∂Ω |∇f|2(q−1)∇(|∇f|2) ≤ 2 Z ∂Ω |∇f|2q: (B.0.11) By trace’sSobolevembeddingtheorem W1,1(Ω) → L1(@Ω), weobtain 2 Z ∂Ω |∇f|2q ≤ c Z Ω |∇f|2q + c Z Ω |∇ 􀀀 |∇f|2q | ≤  Z Ω |∇(|∇f|q)|2 + c() Z Ω |∇f|2q: (B.0.12) The proofiscomplete. In [9], aninterpolationinequalityofEhrling-typeisutilizedtoshowthattheequi-integrability of thefamily R Ω u n 2 (·; t) t∈(0,Tmax) implies theuniformboundednessofsolutions.Herewepresent an interpolationinequalitythatissimilarto[9, Lemma2.1]or[72][Lemma 3.4],andwhichwill be employedtoobtainan Lq estimate with q ≥ 2 for thesolutionsofthesystem(5.2.1). To prove thisinequality,weadapttheargumentusedintheproofofinequality(22)in[6], withsome modifications. Weincludeacompleteproofofthisinterpolationinequalitybelowforthereader’s convenience. Lemma B.0.9. Let Ω ⊂ Rn, with n ≥ 2 be aboundeddomainwithsmoothboundaryand q > n 2 . Then onecanfind C > 0 such thatforeach  > 0, thereexists c = c() > 0 such that Z Ω |w|q+1 ≤  Z Ω |∇w q 2 |2 Z Ω G(|w| n 2 ) 2 n + C Z Ω |w| q+1 + c Z Ω |w| (B.0.13) holds forall w q 2 ∈ W1,2(Ω), and R Ω G(|w| n 2 ) < ∞ where G is continuous,strictlyincreasingand nonnegative in [0;∞) such that lims→∞ G(s) s = ∞. Proof. Wecall (s) = 8>>>>< >>>>: 0 |s| ≤ N 2(|s| − N) N < |s| ≤ 2N |s| |s| > 2N: (B.0.14) One canverifythat Z Ω ||w| − (w)|q+1 ≤ (2N)q Z Ω |w| (B.0.15) 124 and, Z Ω (w) n 2 ≤ N n 2 G(N n 2 ) Z Ω G(|w| n 2 ): (B.0.16) Notice that |∇ ((w)) q 2 |2 ≤ c|w|q−2|∇w|2, forsome c > 0, andcombinewithLemma B.0.2, we obtain Z Ω ((w))q+1 ≤ c Z Ω |∇((w)) q 2 |2 Z Ω (w) n 2 2 n + C Z Ω (w) q+1 ≤ c  N n 2 G(N n 2 ) 2 n Z Ω |∇w q 2 |2 Z Ω G(|w| n 2 ) 2 n + C Z Ω |w| q+1 : (B.0.17) This leadsto Z Ω |w|q+1 ≤ c Z Ω |(w)|q+1 + Z Ω |(w) − |w||q+1  ≤  N n 2 G(N n 2 ) 2 n Z Ω |∇w q 2 |2 Z Ω G(|w| n 2 ) 2 n + C Z Ω |w| q+1 + (2N)q Z Ω |w|: (B.0.18) WefinallycompletetheproofbychoosingNsufficientlylargesuchthat c  N n2 G(N n2 ) 2 n ≤ . The followingLemmaisusefuliniterationproceduretoobtain L∞ bounds from Lq bounds for some q > 1. Lemma B.0.10. Suppose thatthepositivesequences (ak; bk; uk)k≥1 satisfy thefollowingcondi- tions: 8>>>>>>>>>>< >>>>>>>>>>: u k+1 ≤ a k + b k u k ; P∞ k=1 ak = a < ∞; Q∞ k=1 bk = b < ∞; bk ≥ 1; (B.0.19) for all k ∈ N, then supk uk ≤ ab + bu1. Proof. Wehave uk+1 ≤ ak + bkuk ≤ ak + ak−1bk + bkbk−1uk−1 125 ≤ ak + Xk−2 i=0 ak−1−i Yi j=0 bk−j + u1 Yk i=1 bi ≤ b Xk i=1 ai ! + bu1 ≤ ab + bu1: 126 APPENDIX C REGULARITY THEORY C.1 Parabolicregularity In ordertoobtain Lp−Lq estimates forsolutionstoparabolicequations,weneedsomeestimates on theheatsemigroupunderNeumannboundaryconditions.Interestedreadersarereferredto [62][Lemma 1.3]formoredetailsabouttheproof. Lemma C.1.1. Let 􀀀 etΔ  t≥0 be theNeumannheatsemigroupin Ω, andlet 1 > 0 denote thefirst nonzeroeigenvalueof −Δ in Ω under Neumannboundaryconditions.Thenthereexistconstants C1; :::;C4 depending on Ω only whichhavethefollowingproperties. 1. If 1 ≤ q ≤ p ≤ ∞ then etΔw Lp(Ω) ≤ C1  1 + t −n 2 ( 1 q −1 p )  e−λ1t ∥w∥ Lq(Ω) for all t > 0 (C.1.1) holds forall w ∈ Lq(Ω) satisfying R Ω w = 0. 2. If 1 ≤ q ≤ p ≤ ∞ then ∇etΔw Lp(Ω) ≤ C2  1 + t −1 2 −n 2 ( 1 q −1 p )  e−λ1t ∥w∥ Lq(Ω) for all t > 0 (C.1.2) is trueforeach w ∈ Lq(Ω). 3. If 2 ≤ p < ∞ then ∇etΔw Lp(Ω) ≤ C3e−λ1t ∥∇w∥ Lp(Ω) for all t > 0 (C.1.3) is validforall w ∈ W1,p(Ω). 4. Let 1 < q ≤ p < ∞. Then etΔ∇ · w Lp(Ω) ≤ C4  1 + t −1 2 −n 2 ( 1 q −1 p )  e−λ1t ∥w∥ Lq(Ω) for all t > 0 (C.1.4) holds forall w ∈ (C∞ 0 (Ω))n. Consequently,forall t > 0 the operator etΔ∇· possesses a unique determinedextensiontoanoperatorfrom Lq(Ω) into Lp(Ω), withnormcontrolled accordingto (C.1.4). 127 Consequently,wehavethefollowinglemma,whichderivesestimatesonsolutionsoftheparabo- lic equations.Formoredetails,seeLemma 2:1 in [14]. Lemma C.1.2. Let Ω ⊂ Rn, with n ≥ 2 be openboundedwithsmoothboundary, p ≥ 1 and q ≥ 1 satisfy 8>>>>>>< >>>>>>: q < np n−p ; when p n: Assuming that g0 ∈ W1,q(Ω), f ∈ C 􀀀 ¯Ω × [0; T)  , and g ∈ C 􀀀 ¯Ω × [0; T)  ∩ C2,1 􀀀 ¯Ω × (0; T)  ∩ C ([0; T);W1,q(Ω)) is aclassicalsolutiontothefollowingsystem 8>>>>>>< >>>>>>: gt = Δg − g + f in Ω × (0; T); ∂g ∂ν = 0 on @Ω × (0; T); g(·; 0) = g0 in Ω (C.1.5) for some T ∈ (0;∞]. If f ∈ L∞ ((0; T); Lp(Ω)), then g ∈ L∞ ((0; T);W1,q(Ω)). Proof. Wehave g(·; t) = et(Δ−1)g0 + Z t 0 e(t−s)(Δ−1)f(·; s) ds: (C.1.6) Weapply ∇ to bothsidesandmakeuseofLemma C.1.1 to obtainthat ∥∇g(·; t)∥ Lq(Ω) ≤ ∇et(Δ−1)g0 Lq(Ω) + Z t 0 ∇e(t−s)(Δ−1)f(·; s) Lq(Ω) ds ≤ ce−(λ1+1)t ∥∇g0∥ Lq(Ω) + c sup t>0 ∥f(·; t)∥ Lp(Ω) Z ∞ 0 (1 +(t − s) −1 2 −n 2 ( 1 p −1 q ))e−(λ1+1)(t−s) ds (C.1.7) The conditionsof p; q imply that −1 2 − n 2 ( 1 p − 1 q ) > −1, whichmakestheintegralontheright convergent.Therefore,weobtain sup t>0 ∥∇g(·; t)∥ Lq(Ω) ≤ c ∥∇g0∥ Lq(Ω) + c sup t>0 ∥f(·; t)∥ Lp(Ω) ; (C.1.8) which concludestheproof. 128 The followingparabolicregularityresultplaysaimportantroleinthestronglydegeneratecase where infs≥0 D(s) =0. Indeed,itwasprovedthatequation(C.1.5) possessesaglobalbounded solution underasuitableslowgrowthconditionof f. Precisely,wehavethefollowingproposition, which isadirectapplicationofCorollary1.3in[61] with n = 2. PropositionC.1.1. For each a > 0, q >n, K > 0 and  > 0, thereexist C(a; q;K; ) > 0 such that if T ≥ 2 , f ∈ C0(¯Ω×[0; T]), and V ∈ C0(¯Ω×[0; T])∩C2,1(¯Ω ×(0; T))∩C0([0; T);W1,q(Ω)) aresuchthat (C.1.5) is satisfiedwith Z t+τ t Z Ω |f|2 lnα(|f| + e) < K for all t ∈ (0; T −  ): (C.1.9) and ∥V0∥ W1,q(Ω) < K; then |V (x; t)| ≤ C(a; q;K; ) for all (x; t) ∈ Ω × (0; T): (C.1.10) C.2 Regularityforchemotaxissystems In thischapter,weshallapplyMoser-Akikositerationprocedure(see[2, 1]) toobtain L∞ bounds from Lp-bounds forsome p > 1 for variouschemotaxismodelswithhomogenuousNeu- mann boundaryconditionorgeneralnonlinearNeumannboundarycondition. C.2.1 Introduction Weconsiderthefollowingchemotaxissysteminaopen,boundeddomainwithsmoothbound- ary Ω ⊂ Rn, with n ≥ 2 8>>< >>: ut = Δu − ∇ · (u∇v) + f(u) x ∈ Ω; t ∈ (0; Tmax); vt = Δv + u − v x ∈ Ω; t ∈ (0; Tmax); (KS) where f ∈ C([0;∞)) such that f(u) ≤ c(1+up) with c; p ≥ 0 under nonlinearNeumannboundary condition: @u @ = g(u); @v @ = 0; x ∈ @Ω; t ∈ (0; Tmax); (C.2.1) 129 where  is theoutwardnormalvectorand g(u) ≤ cuq with c; q ≥ 0. Thesystem(KS) iscom- plemented withthenonnegativeinitialconditionsin C2+γ(Ω), where ∈ (0; 1), notidentically zero: u(x; 0) = u0(x); v(x; 0) = v0(x); x ∈ Ω; (C.2.2) In thischapter,weassumethat (u; v) is aclassicalsolutionof(KS) withinitialcondition(C.2.2) under nonlinearboundaryconditions(C.2.1) in Ω×(0; Tmax), where Tmax ∈ (0;∞] is themaximal existence time. TheoremC.2.1. If u ∈ L∞ ((0; Tmax);Lr0(Ω)) for some r0 > max n n 2 ; n(p−1) 2 ; n(q − 1) o , then u ∈ L∞ ((0; Tmax);L∞(Ω)). Remark C.2.1. One canalsofollowtheargumentasin[48] toprovetheabovetheorem. Remark C.2.2. The L n 2 +-criterion forhomogeneousNeumannboundaryconditionshasbeenstud- ied in[5] forgeneralchemotaxissystemsandin[67] forthefullyparabolicchemotaxissystem, both withandwithoutalogisticsource.However,Theorem C.2.1 not onlyaddressesnonlinear Neumann boundaryconditions,butalsoemploysadifferentanalysisapproachcomparedto[5, 67]. Insteadofutilizingthesemigroupestimate,theanalysisinTheorem 2.5.1 reliesonthe Lpregularitytheoryforparabolicequations. C.2.2 AnreverseHolder’sinequality WerelyonthefollowingreverseHolder’sinequality,whichisthemilestoneinestablishing Moser-Akikositerationprocedure. Lemma C.2.2. Let (u; v) be aclassicalsolutionof (KS) on (0; Tmax) and Ur := max ( 1; ∥u0∥L∞(Ω); sup t∈(0,Tmax) ∥u(·; t)∥Lr(Ω) ) : If supt∈(0,Tmax) ∥u(·; t)∥Lr(Ω) < ∞ for some r > max n n; n(p−1) 2 ; n(q − 1) o , thenthereexistscon- stant C > 0 independent of r such that U2r ≤ c 1 2r−nk r 3 2rU 1+ (n+2)k 4r−2nk r (C.2.3) 130 where k = max {p − 1; 2q − 2}. Proof. Through outthisproof,thenotation c, unlessbeingspecified,representsapositiveconstant independent of ; r. Multiplying u2r−1 to thefirstequationof(KS), weobtain 1 2r d dt Z Ω u2r = Z Ω u2r−1 [Δu − ∇(u∇v) + f(u)] = −2r − 1 r2 Z Ω |∇ur|2 dx + 2r − 1 r Z Ω ur∇ur · ∇v + Z Ω f(u)u2r−2 + Z ∂Ω g(u)u2r−1 dS; ≤ −2r − 1 r2 Z Ω |∇ur|2 dx + 2r − 1 r Z Ω ur∇ur · ∇v + c Z Ω u2r−1 + c Z Ω u2r−1+p + c Z ∂Ω u2r−1+q dS (C.2.4) By Lemma C.1.2, wededucethat v ∈ L∞ ((0; Tmax);W1,∞(Ω)). This,togetherwithHolder’s inequality implies 2r − 1 r Z Ω ur∇ur · ∇v ≤ c Z Ω ur|∇ur| ≤  Z Ω |∇ur|2 + c  Z Ω u2r: (C.2.5) WemakeuseofTraceSobolev’sEmbeddingTheoremandYoung’sinequalitytohandlethebound- ary integralasfollows: Z Ω u2r+q−1 dS ≤ c Z Ω u2r+q−1 +  Z Ω |∇ur|2 + c  Z Ω u2r+2q−2; (C.2.6) for any  > 0. From(C.2.5) and(C.2.6), wehave d dt Z Ω u2r + Z Ω u2r ≤ 2r  c − 2r − 1 r2 Z Ω |∇ur|2 + cr  + 1  Z Ω u2r + cr Z Ω u2r−1 + cr Z Ω u2r+p−1 + cr Z Ω u2r+q−1 + cr  Z Ω u2r+2q−2 Substituting c = r−1 r2 into this,andnoticingthat r >n ≥ 2, wehave d dt Z Ω u2r + Z Ω u2r ≤ −2 Z Ω |∇ur|2 + cr3 Z Ω u2r + cr Z Ω u2r+p−1 + cr Z Ω u2r−1 + cr Z Ω u2r+q−1 + cr3 Z Ω u2r+2q−2 131 ≤ −2 Z Ω |∇ur|2 + cr3 Z Ω u2r+k + cr3; (C.2.7) where thelastinequalitycomesfrom Z Ω u2r+l = Z u≤1 u2r+l + Z u>1 u2r+l ≤ Z Ω u2r+k + |Ω| (C.2.8) for any l ≤ k. Weset w = ur, andapplyLemma B.0.2 to obtain Z Ω u2r+k = Z Ω w2+k r := Z Ω w¯p ≤ c Z Ω |∇w|2 ¯pa 2 Z Ω w ¯p(1−a) + c Z Ω w ¯p ; (C.2.9) where 2r n > k> −r and ¯p = 2+ k r ; a = 1 − r 2r+k 1 + 1 n − 1 2 = 2n(r + k) (n + 2)(2r + k) < 1: (C.2.10) This impliesthat ¯pa 2 = n(r + k) r(n + 2) < 1 when r > nk 2 : This, togetherwith(C.2.7) andYoung’sinequalityleadsto Z Ω u2r+k ≤ c Z Ω |∇ur|2 ¯pa 2 Z Ω ur ¯p(1−a) + c Z Ω ur ¯p ≤ c Z Ω |∇ur|2 + c −(n+2)r 2r−nk Z Ω ur 2+(n+2)k 2r−nk + c Z Ω ur 2+k r ; (C.2.11) for any  > 0. From(C.2.7) and(C.2.11), weobtain d dt Z Ω u2r + Z Ω u2r ≤ (c − 2) Z Ω |∇ur|2 + cr3 −(n+2)r 2r−nk Z Ω ur 2+(n+2)k 2r−nk + cr3 Z Ω ur 2+k r + cr3; Substituting c = 1 into this,wehave d dt Z Ω u2r + Z Ω u2r ≤ c (n+2)r 2r−nk r3U 2r+r(n+2)k 2r−nk r + cr3U2r+k r + cr3 ≤ c (n+2)r 2r−nk r3U 2r+r(n+2)k 2r−nk r 132 with some c > 1 independent of r. This,togetherwithGronwall’sinequalityimpliesthat Z Ω u2r ≤ max  c (n+2)r 2r−nk r3U 2r+r(n+2)k 2r−nk r ; Z Ω u2r 0 ;  which furtherentails(C.2.3) C.2.3 Proofofmainresults Before provingourmaintheorem,werelyonthefollowinglemma: Lemma C.2.3. Let (u; v) be theclassicalsolutionto (KS) on Ω×(0; Tmax) with maximalexistence time Tmax ∈ (0;∞]. If u ∈ L∞ ((0; Tmax); Lr(Ω)) for some r > max{n 2 ; n(p−1) 2 ; n(q − 1)}, then u ∈ L∞ ((0; Tmax); L2r(Ω)). Proof. If r > max{n 2 ; n(p−1) 2 ; n(q − 1)} > n, thenLemma C.2.2 asserts that u ∈ L∞ ((0; Tmax); L2r(Ω)). Nowwejustneedtoconsidermax{n 2 ; n(p−1) 2 ; n(q − 1)} < r ≤ n. By Lemma C.1.2 we seethat v is in L∞ ((0; T);W1,q(Ω)) for q < rn n−r if r < >: 2n n−2 if n ≥ 3 1 r−1 if n = 2; (C.2.12) and applyHolder’sinequalitytodeducethat Z Ω u2r|∇v|2 ≤ Z Ω u2r+λ  2r 2r+λ Z Ω |∇v| 2(2r+λ) λ  λ 2r+λ : (C.2.13) Since n 2 < r 0 depending on ; r. Substituting c = r−1 r2 into thisyields d dt Z Ω u2r + Z Ω u2r ≤ −2 Z Ω |∇ur|2 + c Z Ω u2r+1 + c Z Ω u2r+λ  2r 2r+λ + c Z Ω u2r + c Z Ω u2r−1 + c Z Ω u2r+p−1 + c Z Ω u2r+q−1 + c Z Ω u2r+2q−2; (C.2.15) with some c > 0 depending on r. UsingGNinequality,thenYoung’sinequalitywith  > 0 and noticing that Ur < ∞, weobtain Z Ω u2r+λ  2r 2r+λ ≤ c Z Ω |∇ur|2 s Z Ω ur 2(1−s) + c Z Ω ur 2 ≤ U2r(1−s) r Z Ω |∇ur|2 s + c()CGNU2r r ≤  Z Ω |∇ur|2 + c(); (C.2.16) where s = 2n(r+λ) (n+2)(2r+λ) ∈ (0; 1). Substituting  = 1 into this,wehave d dt Z Ω u2r + Z Ω u2r ≤ − Z Ω |∇ur|2 + c Z Ω u2r−1 + u2r + u2r+1 + u2r+p−1 + u2r+q−1 + u2r+2q−2 + c; ≤ − Z Ω |∇ur|2 + Z Ω u2r+k + c; (C.2.17) where k = max{p − 1; 2q − 2} and thelastinequalitycomesfrom(C.2.8). Byusing(C.2.11) and noticing that r > kn 2 and Ur < ∞, weobtain Z Ω u2r+k ≤ c Z Ω |∇ur|2 + c(): (C.2.18) From (C.2.17), (C.2.18), wechoose  sufficientlysmalltoobtain d dt Z Ω u2r + Z Ω u2r ≤ c; (C.2.19) 134 with some c > 0. ApplyingGronwall’sinequalitytothis,weobtain sup t∈(0,Tmax) ∥u(·; t)∥ L2r(Ω) ≤ max n ∥u0∥ L∞(Ω) ; c o : The proofiscomplete. ProofofTheorem C.2.1. When r0 > max n n 2 ; n(p−1) 2 ; n(q − 1) o , Lemma C.2.3 deduces that u ∈ L∞ ((0; Tmax);L2r0(Ω)). Thus,wecanassumethat r0 > n. Since r0 > max n n; n(p−1) 2 ; n(q − 1) o , Lemma C.2.2 implies thatthefollowinginequality U2j+1r0 ≤ c 1 2j+1r0 −nk 􀀀 (2jr0)3 1 2j+1r0 U 1+ (n+2)k 2j+2r0 −2nk 2jr0 : for allintegers k ≥ 1. Wetakelogoftheaboveinequalitytoobtain ln U2j+1r0 ≤ aj +  1 + (n + 2)k 2j+2r0 − 2nk  ln U2jr0 ; where aj = lnC 2j+1r0 − nk + 3j ln 2 2j+1r0 + 3 ln r0 2j+1r0 ; bj = 1+ (n + 2)k 2j+2r0 − 2nk : One canverifythat X∞ j=1 aj := A < ∞; and ∞Y j=1 bj := B < ∞: Thus, weobtain U2k+1r ≤ eAUB r0 (C.2.20) for all k ≥ 1. Sending k → ∞ yields U∞ ≤ eAUB r0 : (C.2.21) This assertsthat u ∈ L∞ ((0; Tmax); L∞(Ω)), andthereafterLemma C.1.2 yields that v ∈ L∞ ((0; Tmax);W1,∞(Ω)). 135