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Journal of Functional Analysis
journalhomepage:www.elsevier.com/locate/jfa
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Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities
Matteo Bonfortea,∗, Jørgen Endala,b
aDepartamentodeMatemáticas,UniversidadAutónomadeMadrid(UAM), CampusdeCantoblanco,28049Madrid,Spain
bDepartmentofMathematicalSciences,NorwegianUniversityofScienceand Technology(NTNU),N-7491,Trondheim,Norway
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received30May2022 Accepted16December2022 Availableonline27December2022 CommunicatedbyLuisSilvestre
MSC:
35K55 35K65 35A01 35B45 35R09 35R11
Keywords:
Nonlineardegenerateparabolic equations
Boundednessestimates Greenfunctions
Gagliardo-Nirenberg-Sobolev inequalities
Weestablishboundednessestimatesforsolutionsofgeneral- izedporousmediumequationsoftheform
∂tu + (−L)[um] = 0 in RN× (0, T ),
wherem≥ 1 and−L isalinear,symmetric,andnonnegative operator.The wide classof operators weconsider includes, butisnot limitedto,Lévyoperators.Ourquantitativeesti- matestaketheformofpreciseL1–L∞-smoothingeffectsand absolutebounds,andtheirproofsarebasedontheinterplay betweenadualformulationoftheproblemandestimateson theGreenfunctionof−L andI− L.
Inthelinearcasem= 1,it iswell-knownthattheL1–L∞- smoothingeffect,orultracontractivity,isequivalent toNash inequalities.Thisisalsoequivalenttoheatkernelestimates, whichimplytheGreenfunctionestimatesthatrepresentakey ingredientinourtechniques.
Weestablishasimilarscenariointhenonlinearsettingm> 1.
First,wecanshowthatoperatorsforwhichultracontractivity holds,also provide L1–L∞-smoothing effects inthe nonlin- ear case. The converse implication is not true in general.
A counterexample is given by 0-order Lévy operators like
* Correspondingauthor.
E-mailaddresses:[email protected](M. Bonforte),[email protected](J. Endal).
URLs:http://verso.mat.uam.es/~matteo.bonforte/(M. Bonforte),https://folk.ntnu.no/jorgeen/
(J. Endal).
https://doi.org/10.1016/j.jfa.2022.109831
0022-1236/©2022TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
−L= I−J∗.Theydonotregularizewhenm= 1,butweshow thatsurprisinglyenoughtheydosowhenm> 1,duetothe convexnonlinearity.Thisrevealsastrikingpropertyofnonlin- earequations: thenonlinearityallowsfor betterregularizing properties,almostindependentlyofthelinearoperator.
Finally,weshowthatsmoothingeffects,bothlinearandnon- linear,implyfamiliesofinequalitiesofGagliardo-Nirenberg- Sobolevtype,andweexploreequivalencesbothinthelinear andnonlinearsettingsthroughtheapplicationoftheMoser iteration.
©2022TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Contents
1. Introductionandmainresults . . . . 3
2. Assumptionsandweakdualsolutions . . . . 8
3. Statementsofmainboundednessresults . . . . 11
3.1. L1–L∞-smoothing . . . . 12
3.2. Absolutebounds . . . . 15
3.3. Linearimpliesnonlinear . . . . 15
4. ProofsintheGreenfunctionsetting . . . . 16
4.1. Propertiesofweakdualsolutions . . . . 17
4.2. Reductionargument . . . . 19
4.3. Fundamentalupperbounds . . . . 19
4.4. Boundednessunder(G3) . . . . 27
4.5. Boundednessunder(G1) and(G1) . . . . 33
4.6. Boundednessundercombinationsof(G1) . . . . 37
4.7. Boundednessunder(G2) . . . . 38
4.8. Linearimpliesnonlinear . . . . 38
5. Boundednessresultsfor0-orderoperators . . . . 39
6. SmoothingeffectsVSGagliardo-Nirenberg-Sobolevinequalities . . . . 46
6.1. Thewell-knownlinearcase(m= 1) . . . . 49
6.2. Thenonlinearcase(m> 1) . . . . 53
7. Variousexamples . . . . 61
7.1. Ontheassumption(G1) . . . . 62
7.2. Combinationsofassumption(G1) . . . . 65
7.3. Ontheassumption(G2) . . . . 70
7.4. Ontheassumption(G3) . . . . 73
7.5. Anonexampleofourtheory . . . . 77
Declarationofcompetinginterest . . . . 80
Dataavailability . . . . 80
Acknowledgments . . . . 80
Appendix A. Technicallemmas . . . . 81
Appendix B. L1–L∞-smoothingcontrolsLq–Lp-smoothing . . . . 82
Appendix C. Denselydefined,m-accretive,andDirichletinL1(RN) . . . . 87
C.1. Thesettingofabstractsolutions . . . . 87
C.2. Thesettingofveryweaksolutions . . . . 94
C.3. Comparisonbetweenabstractandveryweaksolutions . . . . 94
Appendix D. Theinverseofadenselydefined,m-accretive,Dirichletoperator . . . . 95
Appendix E. Existenceandaprioriresultsforweakdualsolutions . . . . 98
References . . . . 100
1. Introductionandmain results
Inthispaper,weconsider solutionsofgeneralizedporousmediumequations [115]:
∂tu + (−L)[um] = 0 in QT := RN× (0, T ),
u(·, 0) = u0 on RN, (GPME)
wherem≥ 1,T > 0,0≤ u0∈ L1(RN),andtheoperator−L isatleastlinear,symmetric, nonnegative,1 andincludesLévyoperators2 definedforψ∈ Cc∞(RN) andc≥ 0 as
cψ(x)−
N i,j=1
aij∂x2ixjψ(x)− P.V. ˆ
RN\{0}
ψ(x + z)− ψ(x) dμ(z)
=:Lμ[ψ](x)
(1.1)
where thereal matrix[aij]i,j=1,...,N isnonnegative and symmetric, P.V . isthe Cauchy principalvalue,and:
μ is a nonnegative symmetric Radon measure on RN\ {0} satisfying (Hμ) ˆ
|z|≤1
|z|2dμ(z) + ˆ
|z|>1
1 dμ(z) <∞.
ImportantexamplesaretheLaplacian,thefractional Laplacian,sumofonedimensional fractionalLaplacians,andso-calledconvolutiontypeor0-orderoperatorsgivenas−L= I− J∗ whereJ ≥ 0 satisfiesJL1(RN)= 1.
Boundednessestimatesarethefirst stepon thewayto furtherregularity properties.
Thiswasexploitedine.g.[5] (cf.Theorem2.2in[66]),[57,Section7],[25],[58,Theorem 1.2],[116,Theorem1.2],[59,Theorem1.1],and[38,Theorem1.2].Itisalsoanimportant estimate in obtaining uniqueness in L1 for very weak solutions of (GPME), see e.g.
[39,62,63]. Wewill thereforefocusonsuchestimatesinthis paper.
It is well-known since theworks of Bénilan [11] and Véron [117] thatthe parabolic equation ∂tu− Δ[ϕ(u)] = 0 enjoys L1–L∞-smoothing when ϕ ∈ C1(R) and ϕ(r) ≥ C|r|m−1 (see also [116, Theorem 8.2] in the case of the fractional Laplacian −L = (−Δ)α2,and [114] in thestandardLaplaciancase).Letus thereforefix ϕ(r)=|r|m−1r.
Inthelinearcase(m= 1),thestandardheatequationandthefractionalheat equation still enjoy L1–L∞-smoothing [8,31], but there are cases in which the operator is too
1 Andmoreover,denselydefined,m-accretive,andDirichletinL1(RN).Basically,weneedthecomparison principleandLp-decaytohold forsolutionsof(GPME).Werefer thereadertoAppendixCforfurther information.Notethattheterminology“Dirichletoperator”appearsintheliteraturealsoas“sub-Markovian operator”.Thispropertyisexpressingthefactthattheoperatorhastobeorderpreserving.
2 Thatis,operatorswhicharenonnegativeatanyglobalnonnegativemaximum(usuallycalledthepositive maximumprinciple),seee.g.[51].Whenc> 0,thereis(strong)absorptionin(GPME).
weaktoensuresuchestimates.This cane.g.beseenfortheconvolutiontypeoperators
−L= I− J∗ (cf. [2, Theorem1.4 andLemma1.6]), wherethesolutionsare assmooth as theinitial data. Hence,whenthenonlinearity cannothelp,the operatorneeds tobe strong enoughtoprovide bounded solutions.Oneofourmain concernsis thereforethe following question:
Which operators L produce bounded solutions of (GPME)?
To provide an answerto this intriguing questionwe will extendthe so-calledGreen function method to a wide class of operators. Such a method was developed in a se- ries of papers[14,15,26,27,30,32,33,35] both foroperators on bounded domains and on manifolds,includingtheEuclideanspaceRN.Thekeytoolishavingatdisposalgoodes- timatesforthekernelof(−L)−1,i.e.theGreenfunctionG−L.Now,applyingtheinverse operatoroneachsideofthePDEin(GPME) yieldstheso-calleddual equation
0 = (−L)−1[∂tu] + (−L)−1[(−L)[um]] = G−L∗ ∂tu + um.
Another essentialingredientis theso-calledBénilan-Crandall(time-monotonicity)esti- mate
∂tu(·, t) ≥ − u(·, t)
(m− 1)t inD(RN),
which is a weak version of the fact that the map t → tm1−1u(·,t) is nondecreasing.
Thisiswell-knowntobeaconsequenceofthetime-scalingandcomparisonprinciplefor (GPME), cf. [12,114].3 A combination of the above equations then gives the so-called fundamentalupperboundor “almostrepresentationformula”
um(x0, t)≤ 1
(m− 1)tG−L(· − x0)∗xu(·, t). (1.2) Thelatternameisjustifiedinthesensethattheboundissimilartotheonegivenbythe representationformula(convolutionwiththeheatkernel)inthelinearcasem= 1,where theGreenfunctionG−L(·−x0) isreplacedbytheheatkernelH−L(·−x0) corresponding to the operator.In both cases,the boundedness estimatesfollows directly byapplying various propertiesofG−L(·− x0) andH−L(·− x0).Furtherdetailsontheproofscanbe found inSection4.
Our method allows to recover the well-known L1–L∞-smoothing result, cf. Theo- rem 3.1andFig.1,
3 The estimate is purelynonlinear sinceit degenerates whenm = 1. However,the strongerAronson- Bénilanestimate[4] doholdforthelinearcaseaswell,butitreliesontheoperatoritselfhavingspace-scaling.
Wereferthereadere.g.to[31,Lemma6.1] and[56,p.1270].Thus,theGreenfunctionmethodcanindeed holdforparticularlinearcases.
u(·, t)L∞(RN) t−Nθαu0αθL1α(RN) for a.e. t > 0
where α ∈ (0,2] and θα := (α + N (m− 1))−1, is respectively valid for the Laplacian (−L)= (−Δ) [69,113,114] and thefractional Laplacian−L= (−Δ)α2 [56] (seealso[47, 50] withp= 2).Animmediate consequenceofourapproachisthatalsoLévyoperators L=Lμ with μ comparabletothe measure ofthefractional Laplacianenjoysthesame estimate,seeLemma7.8.Itisalsointerestingtonotethatweareabletotreatoperators whoseGreenfunctionshavedifferentpowerbehaviours.Solutionsof(GPME) withsuch operatorssatisfy
u(·, t)L∞(RN) t−Nθαu0αθL1α(RN)+ t−Nθ2u02θL12(RN) for a.e. t > 0, see Theorem 3.3. Theexamples treatedinSection7.2 are −L= (−Δ)+ (−Δ)α2,rela- tivisticSchrödingertypeoperators−L= (κ2I− Δ)α2− καI withκ> 0,andL beingthe generator of afinite range isotropically symmetric α-stable process inRN with jumps ofsize largerthan 1 removed.Finally,if G−L∈ L1(RN) in (1.2),then weimmediately obtainthefollowing absolutebound(cf.Theorem 3.7andFig.3):
u(·, t)L∞(RN) t−1/(m−1) for a.e. t > 0. (1.3) Alloperators on theform I− L providesuch anestimate (Lemma 7.16),and also the operator−L = (I− Δ)α2 corresponding to the Bessel potential (Lemma 7.21).4 They arefurthermoreexamplesofoperators whichhavebetterboundednesspropertiesinthe nonlinearcasethan inthelinear,seeRemark7.22.
TheGreen functionmethod requires theexistence ofan inverse(−L)−1 with aker- nel G−L satisfying suitable estimates. This of course puts arestriction on the class of operators we are ableto treat. To remedythis fact, we also develop anotherapproach whichconsistsinconsideringGI−Linstead,i.e.,theGreenfunctionassociated withthe resolventoperatorI− L.Inthiscase,theinversealwaysexists,andGI−Lisatleastas goodas G−L. Byrewriting thePDEin(GPME) to∂tu+ (I− L)[um]= um,applying (I− L)−1, and using the time-monotonicity estimate (associated with −L),we obtain thefollowingfundamentalupperbound:
um(x0, t)≤
1
(m− 1)t+u(·, t)mL∞−1(RN)
GI−L(· − x0)∗xu(·, t). (1.4) Hence,we see thatwe have to pay theprice of treatingan equation with thereaction term um, which we then have to reabsorb to be ableto obtaingood estimates in this case.However,note thatwecansplittheestimation of(1.4) intotwocases:
u(·, t)m−1L∞(RN)≤ 1
(m− 1)t and u(·, t)m−1L∞(RN)> 1 (m− 1)t.
4 ThisoperatorcanbewrittenasI− Lμforμ satisfying(Hμ),i.e., ontheform(1.1).
Inthefirstcase,wealreadyhavetheestimateu(·,t)L∞(RN) t−1/(m−1),whileinthe other
um(x0, t)≤ 2u(·, t)m−1L∞(RN)GI−L(· − x0)∗xu(·, t),
from which we can deduce u(·,t)L∞(RN) u0L1(RN) as long as GI−L ∈ Lp(RN) with p∈ (1,∞).Hence,thefundamentalupperbound(1.4) yields
u(·, t)L∞(RN) t−1/(m−1)+u0L1(RN) for a.e. t > 0, (1.5) see Theorem3.5andFig.2.Theoperator−L= N
i=1(−∂x2ixi)α2 provides animportant exampleinthiscasesinceG−L=∞ (forsomevaluesofα),while GI−L∈ Lp(RN). We refer to Lemma7.24 and Remark7.25for furtherinformation.Indeed, this is the first time theGreen functionmethodisableto treatthisoperator.Weendthispartbyalso mentioning thatLévyoperators L=Lμ with μ such that,forα∈ (0,2) andconstants C1,C2,C3> 0,
C1
|z|N +α1|z|≤1≤ dμ
dz(z)≤ C2
|z|N +α1|z|≤1 and dμ
dz(z)≤ C31|z|>1, (1.6) fall intothiscase(Lemma7.26).Thelatterfitswiththe“usual impression”inthePDE communityregardingtheleastassumptionsexpectedonnonlocaloperatorswhichwould produce bounded solutions of (GPME). Nevertheless,we were not able to find such a resultother placesintheliterature.
An alternative to the Green function method is thenowadays standard Moseriter- ation [95,96], which requires the quadratic form associated to the operator to satisfy Gagliardo-Nirenberg-Sobolev (GNS) and Stroock-Varopoulos inequalities. In the case
−L = (−Δ)α2, we refer to [56]. We devote Section 6 to a further discussion on the connectionsbetweenGreenfunctionestimates,heatkernelestimates,andfunctionalin- equalities like GNS.Inthe linearcase m= 1, itis well-known thatL1–L∞-smoothing is equivalent with Nash inequalities (a subfamily of GNS) [97], and moreover, equiv- alent with on-diagonal heat kernel H−L estimates. We present those connections in Theorem 6.1, where wealsoinclude—maybethe less-known—equivalence withSobolev inequalities. Since we are interested inGreen function estimates, we finally provethat theboundG−L |x|−(N−α)impliestheSobolevinequality.IftheGreenfunctionexists, itis givenby
G−L(x) = ˆ∞
0
H−L(x, t) dt.
Hence,off-diagonalheatkernelboundsisneededtogiveestimatesontheGreenfunction.
Inotherwords,weneedmoreinformation onH−Lthanwhatthepreviousequivalences
giveus.Thelinearpanoramaismoreorlesssettled,andwemoveontothenonlinearcase m> 1.Again,L1–L∞-smoothingisequivalentwithafamilyofGNSinequalitieswhichis nowsubcriticalsincem> 1.Thelatterissomehowinterestinginthesensethatweneed aweaker inequality, compared to the linear case, inorder to prove L1–L∞-smoothing throughtheMoseriteration.However,thisinequalityisstillequivalentwiththeSobolev inequalityby [7]. This is incontrast to the absolute bound which is equivalent to the Poincaréinequality!Ingeneral,thelatterinequalitycanonlygiveLq–Lp-smoothinges- timatesthroughaniterationapproach[75,76],andsomehowtheGreenfunctionmethod thenprovidesanimprovementhere(sinceweindeedreachL∞-estimates,cf.(1.3)).See theFigs.4and5forthevarious connections.
Theresolventapproachalsooffersfurtherinterestinginsight.Since
GI−L(x) = ˆ∞
0
HI−L(x, t) dt = ˆ∞
0
e−tH−L(x, t) dt,
evenpooron-diagonalheatkernelboundsforH−LwillgiveGI−L∈ Lp(RN).Thishasat leasttwo consequences:(i) Such estimatesfor H−L imply both Greenfunction bounds and also GNS inequalities, which furthermore imply that solutions of (GPME) (with m > 1) are bounded whenever the Green function method and/or Moser iteration go through.(ii)Iftheoperatorissuchthatsolutionsof(GPME) withm= 1 arebounded, then alsosolutionsof (GPME) withm> 1 are bounded (Theorem3.9).The last item correspondstothe“usualimpression”inthePDEcommunity,butagainwewerenotable tofindagoodreferenceforsuchastatement.Thefirstitemprovidesaclearconnection betweentheGreenfunctionmethodandtheMoseriteration,buttherearesomerather simpleon-diagonalboundsforwhichthealgebraoftheMoseriterationis hardtowork out,while the Green function approachis morestraightforward. Considerfor example H−L(x,t) t−N/αetwhichcorrespondstotheLévyoperatorLμwithμ satisfying(1.6).
It is clear that the linear casehas the estimate u(·,t)L∞(RN) t−N/αetu0L1(RN), for a.e.t > 0, butthe unclear nonlinear caseis in facteasily handledwith theGreen functionmethod.
Wehavethenreachedourfinaltask:
Can the nonlinear case provide bounded solutions in cases when the linear cannot?
Thequestionhasparallelstootherequationsforwhichregularizingeffectsonlyhappen when the nonlinearity is strong enough. Take e.g. the scalar conservation law ∂tu+ div[f (u)] = 0. If f (r) = r, we are in the setting of the transport equation, and the solutions are as smooth as the initial data. Hence, the operator itself is not able to provide smoothing estimates.In thementioned case,f needs to be so-called genuinely nonlineartoprovideregularizingeffects.Asufficientconditionisf(r)> 0 whenN = 1, andf : RN → RN definedasf (r)= (u2/2,u3/3,. . . ,uN +1/N + 1) whenN > 1.L1–L∞- smoothingcanthenbefoundin[104],whileotherregularizingpropertiesine.g.[53].In
this context,we also mention[1] whichtreatse.g.∂tu+ div[f (u)]− Δ[um]= 0. Under some conditions on f , it is proven that properties like boundedness hold whenever it holds for∂tu− Δ[um]= 0.
We alsofound theanswerto theabovequestionbylooking at operators whichwere too weak to provide boundedness estimates by themselves: the family −L = I − J∗, mentioned earlier.Basically,the porousmediumnonlinearity isso strongthatwe were even able to prove that solutions of (GPME) with those operators are bounded as in (1.5). Theorem 5.1 provides the rigorous statement,and what is interesting to note is thattheproofresemblestheGreen functionoftheresolventoperatormethod.
Notation. Derivativesare denotedby , dtd, and ∂xi. We usestandard notationfor Lp, Wp,q, and Cb. Moreover, Cc∞ is the space of smooth functions with compact sup- port, Cb∞ the space of smooth functions with bounded derivatives of all orders, and C([0,T ];Lploc(RN)) the space of measurable functions ψ : [0,T ]→ Lploc(RN) such that ψ(t)∈ Lploc(RN) forallt∈ [0,T ],supt∈[0,T ]ψ(t)Lp(K)<∞,andψ(t)− ψ(s)Lp(K)→ 0 when t → s for all compact K ⊂ RN and t,s ∈ [0,T ]. In a similar way we also define C([0,T ];Lp(RN)). Note that the notion of RN × (0,T ) (x,t) → ψ(x,t) ∈ C([0,T ];Lploc(RN)) is asubtle one.In fact, we mean that ψ has an a.e.-version which is continuous [0,T ] → Lp(RN). Let f,g be positive functions. The notation f g or f g translatesto f ≤ Cg orf ≥ Cg forsomeconstantC > 0. Hence,f g is exactly that f g and f g holdsimultaneously.For α∈ (0,2] and p∈ [1,∞), thequantity (αp+N (m−1))−1willeitherbedenotedbyθporθα,whenthereisnoambiguity.Finally, thefollowing Younginequalityisrepeatedlyused throughoutthepaper:
ab≤ 1
ϑaϑ+ϑ− 1
ϑ bϑ−1ϑ , where a, b > 0 and ϑ > 1. (1.7) 2. Assumptionsandweakdualsolutions
ThespatialdimensionisfixedtobeN ≥ 3,5andtheassumptionsonthedata(u0,m) are6:
5 ThecasesN = 1 andN = 2 aredifferentandcouldfalloutofourgeneralsetting.Anexampleisthe fractionalLaplacian,wheretheconditionN > α playsanessentialroleintheformoftheGreenfunction.
In thiscasewecouldconsiderN ≥ 1, undertheextraconditionα∈ (0,1). Also,theGreenfunctionof the standardLaplacianissign-changingwhenN = 2,anditfallsoutofoursetting asitfailstosatisfy assumption (HG) below.Since we are dealing withSobolev-type inequalities andtheir connection with smoothingeffects,itisworthrecallingthatthoseinequalitiestendtobedifferentindimension1and2:For instance,functionsofH1(R) areautomaticallybounded.
6 Forthepurposeofboundednessresults,thereisnolossofgeneralityinassumingnonnegativeinitialdata.
First, forsign-changingsolutions,thenonlinearityumhastobereplacedby|u|m−1u.Asaconsequence,
−u is a solution of (GPME)whenever u is. Second, considerthe sign-changing solution u with initial datau0,andalsothetwoothernonnegativesolutionsu+andu− correspondingrespectivelytotheinitial data u+0 = max{u0,0} and u−0 = −min{u0,0}.By thecomparisonprinciple,u0 ≤ u+0 impliesu ≤ u+ and −u0 ≤ u−0 implies−u ≤ u−. Wecan combinethe inequalitiestoobtain −u− ≤ u≤ u+, thatis
|u|≤ max{u+,u−} ≤ u++ u−. Also,beingnonnegative,u+ andu− satisfy(someformof) smoothing effectestimate,thatwecansumuptoobtainthesameestimatefor|u|, sinceu+0 andu−0 havedisjoint support.
0≤ u0∈ L1(RN) (see footnote6). (Hu0) The nonlinearity is r→ rm for some fixed m > 1. (Hm) WewillmakerepeateduseoftheGreenfunctions(orfundamentalsolutionsorpoten- tialkernels)G−Lx0 and GxI−L0 ofthe nonnegativeoperator−L and thepositive operator I− L.Acrucial assumptionthroughout thepaper istherefore:
For the operator A, there exists a function GAx0 ∈ L1loc(RN) such that: (HG) 0≤ GAx0= GA0(· − x0) = G0A(x0− ·) a.e. in RN and A[GAx0] = δx0 inD(RN).
Remark2.1.The assumption GxA0 = G0A(·− x0) (possibly)excludesx-dependent oper- ators.To includex-dependent operators, onewouldinstead needGAx0 = GA0(·,x0) and G0A(·,x0) continuousinRN\ {x0}.Inthiscase,A−1[f ] cannotbe writtenas aconvolu- tion,butother thanthat,theproofsgothroughasbefore.
AppendixDprovidesaguideforchecking(HG) forspecificoperators.Letusjustmen- tionthatwhenA isoftheform(1.1) withameasureμ satisfying(Hμ),thenGAx0 satisfy theaboveunder(possibly)someadditionalpropertiesontheheatkernelassociatedwith A.Moreover,foreachsuchoperatorA,wehaveA−1 definedas
A−1[f ](y) :=
ˆ
RN
GyAf = ˆ
RN
G0A(· − y)f = G0A∗ f(y) = GAy ∗ f,
wheneverthatintegralisconvergent.TheGreenfunctionsthatwillbeusedinthispaper satisfy(withCp,K1,K2,K3,C1> 0 allindependentofx0)oneofthefollowingadditional assumptions:
For all R > 0, some x0∈ RN, and some α∈ (0, 2], (G1)
´
BR(x0)G−Lx0 (x) dx≤ K1Rα,
and for a.e. x∈ RN\ BR(x0), G−Lx0 (x)≤ K2R−(N−α).
For all R > 0, some x0∈ RN, and some α∈ (0, 2], (G1)
´
BR(x0)G−Lx0 (x) dx≤ K1Rα,
and for a.e. x∈ RN\ BR(x0), G−Lx0 (x)≤ max{K3, K2R−(N−α)}.
For some x0∈ RN, (G2)
G−Lx0 L1(RN)=G−L0 L1(RN)≤ C1<∞.
For some x0∈ RN and some p∈ (1, ∞), (G3)
GIx−L0 Lp(RN)=G0I−LLp(RN)≤ Cp<∞.
Remark2.2.
(a) Note that there is no ambiguity in assumption (G1). Indeed, we cannot consider Green functions which are merely bounded around x0 since this would contradict theintegrabilitycondition.
(b) Wecanviewassumption(G3) intwoways:(i)Wethinkof−L→ I − L in(GPME), i.e., c= 1 in (1.1).(ii)We thinkof−L in(GPME),butwe wanttouse theGreen functionoftheresolventofthatoperator,i.e.,c= 0 in(1.1).Inboth cases,if−L is suchthatthe correspondingheat equation givesL1-decay,thenG0I−LL1(RN)≤ 1 (seeLemma7.16).Hence,ifweconsideritem (i),weareactuallyinthecase(G2).
(c) Fornow,wejustremarkthatthefractionalLaplacian/Laplacian−L= (−Δ)α2 with α∈ (0,2] satisfy(HG) and(G1)–(G3),while −L ofthefull form(1.1) (with c> 0) satisfies(HG) and(G2).OtherimportantexamplescanbefoundinSection7,where heatkernelboundsandFouriermethodsareusedtoobtainGreenfunctionbounds.
Ifwe applytheinverse(−L)−1 oneachsideof thePDEin(GPME),we get 0 = (−L)−1[∂tu] + (−L)−1[(−L)[um]] = ∂t
Gx−L0 ∗xu) + um.
Wethusdefine asuitableclassofsolutionsasthefollowing:
Definition 2.1 (Weak dual solution).We say thatanonnegative measurable function u is aweakdual solution of(GPME) if:
(i) u∈ C([0,T ];L1(RN)) andum∈ L1((0,T );L1loc(RN)).
(ii) For a.e.0< τ1≤ τ2≤ T ,and allψ∈ Cc1([τ1,τ2];L∞c (RN)),
τ2
ˆ
τ1
ˆ
RN
(−L)−1[u]∂tψ− umψ dx dt
= ˆ
RN
(−L)−1[u(·, τ2)](x)ψ(x, τ2) dx− ˆ
RN
(−L)−1[u(·, τ1)](x)ψ(x, τ1) dx.
(2.1)
(iii) u(·,0)= u0 a.e.inRN. Remark2.3.
(a) We need to argue that(−L)−1[u] ∈ C([0,T ];L1loc(RN)) in order to make sense of theabovedefinition.Byusing(G1) and(G1),wehave
(−L)−1[1Br(x0)](x) = ˆ
Br(x0)
G−Lx0 (x) dx≤ C
whichimpliesthat ˆ
RN
(−L)−1[u(·, t)](x)1Br(x0)(x) dx≤ Cu(·, t)L1(RN)
for allr > 0 and all x0 ∈ RN. This makesus ableto complete theargument as in Remark2.1in[15].Inthecaseof (G2),wegetthestronger
ˆ
RN
(−L)−1[u(·, t)](x) dx ≤ Gx−L0 L1(RN)u(·, t)L1(RN),
and hence,(−L)−1[u]∈ C([0,T ];L1(RN)).
(b) Later,wewillalsousethe weak dualformulationfor
∂tu + (I− L)[um] = um ⇐⇒ ∂tu− L[um] = 0.
Part(ii)of theabovedefinitionthenlookslike
τ2
ˆ
τ1
ˆ
RN
(I− L)−1[u]∂tψ− umψ + um(I− L)−1[ψ] dx dt
= ˆ
RN
(I− L)−1[u(·, τ2)](x)ψ(x, τ2) dx− ˆ
RN
(I− L)−1[u(·, τ1)](x)ψ(x, τ1) dx.
Weagainneed(I− L)−1[u]∈ C([0,T ];L1loc(RN)).Since ˆ
RN
(I− L)−1[u(·, t)](x) dx ≤ G0I−LL1(RN)u(·, t)L1(RN),
whichisfinitebyRemark2.2,wegetthestronger(I− L)−1[u]∈ C([0,T ];L1(RN)).
(c) Regarding uniqueness and very weak solutions. In many cases, weak dual so- lutions are very weak in the sense of [62,63]. For instance this happens when Cc∞ ⊂ dom(−L).A simple,and yettechnical, proof followsby approximatingL[φ]
byasequenceψnofadmissibletestfunctionsin(2.1).Asaconsequence,wecanuse theresultsof[62,63] toconcludeexistenceanduniquenessofweakdualsolutionsin L1(RN) sincewewillshowthattheyareaprioribounded.Ageneralexistenceresult forourpurposes canbefound inProposition4.1.
3. Statementsofmainboundednessresults
Wepresent someexplicitestimates regardinginstantaneousboundedness whichrely onthe assumptions (G1)–(G3). Allof ourresults originatefrom what is oftenreferred
N
i=1(−∂2xixi)α2
(−Δ)α2
Lμsuch that
|z|−(N+α) dμdz |z|−(N+α)
u(·, t)L∞ t−Nθαu0αθL1α Sobolev
Fig. 1. Operators that fall into the setting of Theorem 3.1, see Section 7. Note that the operator N
i=1(−∂2xixi)α2 actuallyenjoysTheorem3.5,butafterascalingargument,wecandeducethebetteresti- mateabove(Remark7.25).AccordingtoSection6theyshouldfurthermoreenjoyaSobolevinequality.
to as fundamentalupper bounds, see Theorem 4.6in Section4. These bounds provide an “almostrepresentationformula”similartotheonegivenbyconvolutioninthelinear case(m= 1).
3.1. L1–L∞-smoothing
We start with the assumptions (G1) and (G1) which impose the moststructure. In effect,wededucewell-knownresults.
Theorem3.1(L1–L∞-smoothing).Assume(Hu0)–(HG),andletu beaweakdualsolution of (GPME) with initialdatau0.
(a) If(G1) holds,then
u(·, t)L∞(RN)≤ C(m, α, N )
tN θα u0αθL1α(RN) for a.e. t > 0, where θα:= (α + N (m− 1))−1,C(m):= 2m−1m ,and
C(m, α, N ) := 2m1C(m)N θα
m
m− 1
αθα
K1(N−α)θαK2αθα. (b) If (G1) holds,then
u(·, t)L∞(RN)≤
⎧⎨
⎩
C(m,α,N )
tN θα u0αθL1α(RN) if 0 < t≤ t0 a.e.,
C(m)˜
tm1 u0Lm11(RN) if t > t0 a.e., where C(m)˜ := (2m(m− 1)−1C(m)K3)1/m and
t0:= 2m m m− 1
−(m−1)
C(m)K1mK
m−1αm
2 K−(
m−1αm+(m−1))
3 u0−(m−1)L1(RN).