一种非线性抛物方程在一般几何流下的梯度估计.pdf

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数学杂志
VoN詈2022).(PRC)
GRADIENTESTIMATEFORANONLINEAR
PARABOLICEQUATIONUNDERGEOMETRICFLOW
WUMeng-fei
(SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,China)
Abstract:Inthispaper,throughtheLi-YaugradientestimateandJunSun'sresearchon
thegradientestimateofheatequationundergeneralgeometricflow,wewillderivelocalgradient
estimatesforpositivesolutionsofanonlinearparabolicequationonRiemannianmanifoldunder
'
sametime,wegiveacorrespondingHarnackinequality.
Keywords:gradientestimate;geometricflow;anonlinearparabolicequation;HarnackinR
equality
2010MRSubjectClassification:53C44;53C21
Documentcode:AArticleID:0255-7797(2022)04-0287-13
1Introduction
StartingwiththeclassicalworkofLiandYau[1],LiandYauprovedthecelebrated
Li-Yaugradientboundforpositivesolutionsoftheheatequationandobtainedtheclassical
Harnackinequlity.
Let)beann-
oftheheatequation
du八
石=Au,(1-1)
wesupposeRic>'K,whereK>
solutionof()satisfies
呼—Q<na2Kna2
-------------------1----------Vq>1.
u2u2(a—1)2t'
InthespecialcasewhereRic>0,onehastheoptimalLi-Yaubound
na2222
|Vu|2ut
—a—"2PVq>1.
u2u
Thegradientestimateisanimportanttoolinthestudyofellipticandparabolictype
-Yaugradientboundshavebeenstudiedbymanyresearchers.
*Receiveddate:2021-03-24Accepteddate:2021-05-25
Biography:WuMengfei(1996-),male,bornatNanyang,Henan,postgraduate,majoringeometric
-mail:******@:.

Hamilton[2]discoveredamatrixLi-
Li-YauboundunderweakerconditionswassubsequentlyestablishedbyCao-Ni[3]onKahler
,[4]provedagradientestimateforpositivesolutionsoftheheat
[5]generalizedLiu'sresultstoageneralgeometricflow.
Inrecentyears,therearemoreandmoreresearchesonthegradientestimatesforpositive
solutionstosomenonlinearparabolicequations.
Forexemple,in2009,Lu,Ni,VazquezandVillanistudiedtheporousmediumequation
(PMEforshort)
dtu=Aum()
onmanifolds[6].TheygotalocalAronson-,Huangand
Liin[7]generalizedtheresultsofLu,Ni,VazquezandVillani[6]onthePMEandobtained
Li-Yautype,HamiltontypeandLi-,XieandZhou[8]
haduniformlypromotedtheseresultstotheRicciflow.
Inthispaper,wefocusonthegradientestimatesofpositivesolutionstoanextremely
[9],who
consideredalocalgradientestimateofpositivesolutionsforthefollowingparabolicequation
Au+aulogu+bu=0inMn,
wherea,b6Rareconstantsforcompletenoncompactmanifoldswithafixedmetricand
curvaturelocallyboundedbelow.
In[10],YanggeneralizedMa'sresult[9]andderivedalocalgradientestimatesfor
positivesolutionstotheequation
du
=Au+aulogu+bu,
~dt
wherea,b6Rareconstantsforcompletenoncompactmanifoldswithafixedmetricand
curvaturelocallyboundedbelow.
Replacingubyeb/au,theequationbecomes
ut=Au+aulogu.()
In2020,[11]investigatedgradientestimatesforpositive
solutionsto()
ofLi-Yau,,HamiltonandLi-Xutoamoregeneralnonlinearparabolicequationalong
theRicciflow.
Inthispaper,wewillfollowclosely[11]andderivelocalgradientestimatesforpositive
solutionsof()
regardedasagenerlizationofWang'
asfollows:
=2hij'a4):.

wherehjisasecond-
whichwasintroducedbyHamilton[12]in1982.
Tostateourmainrsult,weintroducethreeC1functions卩(t)and7(t):(0,+x)t
(0,+x).SupposethatthreeC1functionssatisfythefollowingconditions:
(C1)a(t)>1.
(C2)a(t)and卩(t)satisfythefollowingsystem
[乎-2必12(警_a')2'
\晋>0,
l吟+20-
(C3)y(t)satisfies十—(手—az)a0.
(C4)Y(t)isnon-decreasing,anda(t)isalsonon-decreasingandboundeduniformly.
Herea'L箫/0L警andY'Ld-
Detailedcalculationofsomespecificfunctionsa(t),卩(t)and7(f)canbefoundin[11].
Westateourresultsasfollows.
MainTheorem1Let(Mn,g(t))tE^QT]beasmoothoneparameterfamilyofcomM
pleteRiemannianmanifoldsevolvingby()fortinsometimeinterval[0,T].LetMbe
completeundertheinitialmetricg(0).Supposethatthereexistthreefunctionsa(t),甲(t)
andY(t)whichsatisfytheaboveconditions(C1),(C2)(C3)and(C4).Givenxo6Mand
R>0,letubeapositivesolutiontotheequation()dtu—Au+auloguinthecube
:—{(x,t)|d(x,xo,t)£2R,0£t£T},
existconstantsKi,K2,K3,K4>0suchthat
Ric>'Kig,K2q<h<K3g,|Vh|<K4()
4
onQ2r,(x,t)6Qr,t,wehaveifOOi£CiforsomeconstantCi,then
|Vu|2ut2(1a/Ki\n2C
—+aalogu-Ca(R2+a+K2丿+R27
u2
33no?Kla/3(Ki+K2)na2
+42(a—1)3+a—1
+an2(K2+K3)+a2n292+a甲,
whereC—C(n,Ci)£C2forsomeconstantC2,then
|Vu|2—aVut+aailoguCa小2((R12+〃〒Ki+a+K2\丿n2Ca4
u2
33na2K(//3(Ki+K2)na2
+43(a-1)1+a-1
+an2(K2+K3)+a2n292+ap,:.

whereC=C(n,C2)isaconstant.
(Mn,g(t))te[0t]beasmoothoneparameterfamilyofcomplete
Riemannianmanifoldsevolvingby()fortinsometimeinterval[0,T].LetMbecomplete
undertheinitialmetricg(0).SupposethatthereexistconstantsKi,K2,K^,K4>0suchthat
Ric>Kig,-K^g<h<K^g,|Vh|<K4,Givenxo6MandR>0,letubeapositive
solutiontotheequation()inthecubeB^r,t5={(x,t)|d(x,xo,t)C2R,0tT}.
Thenthefollowingspecialestimatesarevalid.
-Yautype
/、annKia2
a(t)=constant,旳=—+,Y(t)=护with0<0C2.
aL1
匚厂|Vu|2—auVt+aa、logucCa2((R12+丁+aa'21R12+a+
33na2K(J3(K〔+K2)na2
+43(a-1)1+a-1
+an2(K2+K3)+a2n2甲2+ap

a(t)=e2Kt,p(t)=⑴二址冰讥
^2——+aaloguCCa2(穆++a+
u2uR2R
+Ca433na2K(y/3(Ki+K2)na2
+R2te2Kit+43(a-1)3+0'1
+an2(K2+K3)+a2n2p2+ap.
-Xutype
sinh(Kit)cosh(Kit)-Kit
a(t)=1+p(t)=2nKi[1+coth(Kit)],7(f)=tanh(Kit).
sinh2(Kit)
|Vu|2ut、*2(1VK
------2a+aaloguCCaI-=72+---------------------------+a+
u2------------u----------------------------R2R
C33na2K43a/3(Ki+K2)na2
+R2tanh(Kit)+42(a一1)1+a—1
+an2(K2+K3)+a2n2p2+ap.
-Xutype
n1
a(t)=1+2Kit,p(t)=—+nKi(1+2Kit+“Kit),Y(t)=Kitwith"24-:.

|Vu|2ut12(1vK
-------2----------a---------+aalogu^Ca(+--------r--------+a+
Ca433na2K(y/3(Ki+K2)no?
+R2Kit+43(a—1)3+a—1
+an2(K2+K3)+a2n2甲2+ap
Thelocalestimatesaboveimplyglobalestimates.
(Mn,g(0))beacompletenoncompactRiemannianmanifoldwithM
outboundaryandg(t)evolvingby().SupposethatthereexistconstantsK1;K2,K3,K4>
0suchthatRic>'K\g,Kqg<h<K3g,|Vh|<(x,t)beapositivesolutionto
theequation().For(x,t)6Mnx(0,T],then
|Vu|233na2K(+J3(Ki+K2)na2
—a—+aaloguCq2(Ki+a)++a_1
u243(a—1)3
+an2(K2+K3)+a2n2p2+ap.
Asaconsequenceofthegradientestimate,wecanobtainthefollowingHarnackinequlity.
(Mn,g(0))beacompletenoncompactRiemannianmanifoldwithM
(t)evolvesby().Supposethat
thereexistconstantsK1;K2,K3,K4>0suchthatRic>K〔g,Kqg<h<K3g,|Vh|<
(x,t)beapositivesolutiontotheequation().Thenforall(x〔'tj6Mnx(0,T)
and(X2,t2)6Mnx(0,T)suchthatti<t2,wehave
u(X2,t2)
1|Y'(s)|4t2()
^^dt+[[Q+|alogN|]dt),
£u(x1;ti)+
o2(t2—ti)titi
whereN=maxMnx[o;T]u,and
Q
1小2E\33na2K43a/3(Ki+K2)na2丄”
Ca2(Ki+a)+-----------+---------------------------------------------1+an2(K2+K3)+a2n2p2+ap.
a(t)43(a—1)=3fa1
2ProofofMainTheorems
ToproveTheorem1,/=
(a—dt)f=|Vf|2—af.()
LetF=|Vf|2—aft+aaf—ap,wherea=a(t)andp=p(t).Then
Af=ftaf|Vf|2=—F9(Rf|2—p.()
aa:.

([5],Lemma3)Supposethemetricevolvesby().Then,foranysmooth
functionf,wehaved|Vf|2=-2h(Vf,Vf)+2(Vf,f,and磊Af=A盍f-2〈h,V2f〉-
2(divh—*V(trgh),Vf〉.Here,divhisthedivergenceofh.
(M,g(t))
(A-dt)F2f++(2P-aj土F-a2n(K2+K3)2-3V^aK4|Vf|
-2(Ki+K2)|Vf|2-2VfVF+2a(a-1)|Vf|2+aaAf.
.
AF
=A|Vf|2-aAft)+aaAf
=2|fijf+2fjfiij+2Rijfifj-a(Af)t-2a〈h,Vf〉一2a〈divh一V(trgh),Vf〉+aaAf
=2|fijF+2fjfiij+2Rijfifj-a(ft一af-|Vf|2)t-2a〈h,V2f〉
—2a(divh—2V(tr3h),Vf〉+aaAf,
()
and
dtF=(|Vf|2)t-aftt-aft+azaf+aaft-apz-azp
()
=2VfV(ft)一2hijfifj—aftt一aft+a'af+aaft—apZ一a'p.
Wefollowthatfrom()and()
(A一dt)F=2|fij〔2+2fjfiij+2Rijfifj—a(ft—af—|Vf|2)t—2a〈h,V2f〉
一2a〈divh一V(trgh),Vf〉+aaAf+2hijfifj
一2VfV(ft)+aftt+aft—a'af—aaft+aP‘+a‘P
=2|fij〔2+2fjfiij+2Rijfifj—a(|Vf|2)t—2a〈h,Vf〉
一2a〈divh一V(trgh),Vf〉+aaAf+2hijfifj
-2VfV(ft)+aft-a'af+apz+a'p
()
=2|fijf+---2a〈h,Vf〉一2a〈divh一V(trgh)'Vf〉+aaAf
+2Rijfifj+2hijfifj—2ahijfifj+2VfV(Af)+2aVfV(ft)
-2VfV(ft)+aft-a'af+apz+a'p
=2|fijf+---2a〈h,Vf〉一2a〈divh一V(trgh),Vf〉+aaAf
+2Rijfifj+2hijfifj—2ahijfifj—2VfVF+2a(a一1)|Vf|2
aft—a'af+apz+a':.

Ourassumptionimpliesthat
|h|2<(K2+K3)2|g|2—n(K2+K3)2.
ApplyingthoseboundsandYoung'sinequalityyields
|a〈h,<2f〉|<||V2f|2+1a2|h|2<||V2f|2+2a2n(K2+K3).
Ontheotherhand,
1…1..33「
divh—V(trsh)—屮Vihji-屮V;hij<|g||Vh|<2^X4.
Therefore,wearriveat
(A-dt)F2|fj|2-a2n(K2+K3)2-30aK4Vf+aaAf
+2Rijfifj+2hijfifj—2ahjfifj—2VfVF+2a(a—1)|V/|2()
azft—a'af+apz+azp.
Further,byutilizingtheunitmatrix(6j)nxnand(),weobtain
(A一dt)FL|fij+|+|Vf|2-ft—af
—a2n(K2+K3)2-^/naK4|Vf|+2Rijfifj+2hijfifj—2VfVF
2
+2a(a—1)|Vf|2+aaAf+L
n
L|fij+n〃ij|+|Vf|2一ft—af
一a2n(K2+K3)2-3V^aK4|Vf|+2Rijfifj+2hijfifj—2VfVF
+2a(a—1)|Vf|2+aaAf
2|fij+n^ij|+aF一(K2+K3)2—3V^aK4|Vf|
—2(Ki+K2)|Vf|2—2VfVF+2a(a—1)|Vf|2+aaAf.
Thisfinishestheproofofthelemma.
ProofoftheMainTheorem1LetG—7(t)FandY(t)>0benon-decreasing.
Then
(A-dt)GL7(A-dt)F-7zF
27|fij+ngij|+1G—7a2n(K2+K3)2—3了/^aK4|Vf|
a
—2y(Ki+K2)|Vf|2—2VfVG+2a7(a—1)|Vf|2+ayaAf—/F
L79|27G—ya2n(K2+K3)2—3^^aK4|Vf|
fij+ngj|+
7.
—2y(Ki+K2)|Vf|2—2VfVG+2ay(a—1)|Vf|2+ayaAf.
():.

Byourassumptionoftheboundsofhandtheevolutionofthemetric,weknowthatg(t)is
uniformlyequivalenttotheinitialmetricg(0),thatis,
e-2K2Tg(0)<g(t)<e2K3Tg(0).
Thusweknowthat(M,g(t))isalsocompletefort6[0,T].Nowlet讽r)beaC2function
on[0,+x)suchthat
1ifr6[0,1],
讽r)=
0ifr6[2,+x),
M'(r)|2
0<讽r)<1,训(r)<0,妙"(r)>—C,<C,
讽r)

d(x,xo,t)
0(x,t)=0(d(x,xo,t))=妙
R
wherep(x,t)=d(x,xo,t).Forthepurposeofapplyingthemaximumprinciple,thearguM
mentof[Calabi1958]allowsustoassumethatthefunction0(x,t),withsupportinQ2r,t,
isC2atthemaximumpoint.
Forany0<Ti<T,let(xi,ti)bethepointinQ2R,Ti,atwhich0Gachievesits
,becauseintheothercasethe
(x,0)=0,weknowthatti>(xi,ti),wehave
V(0G)=FV0+0VG=0,A(0G)<0,dt(0G)>0.()
Therefore,
0>(A—dt)(0G)=(A0)G—0tG+0(A—dt)G+2V0-VG.()
UsingtheLaplaciancomparisontheorem,wehave
A0=汐AP+“〃搭>-R
⑵11)
Furthermore,wehave
|V0|2=⑷2|Vp|2<C
()
0=妙R2—R2.
Byourassumption,G(xi,ti)>
geometricflow[Hamilton1995a],wecalculateatthepoint(xi,ti)
RJRJh(S,S)dsG
()
>叫R)RK2PG>—WK2G,:.

where7^isthegeodesicconnectingxandx°underthemetricg(ti),Sistheunitetangent
vectorto7^anddsistheelementofarclength.
Allthefollowingcomputationsareatthepoint(xi,ti).Wehave
2
fij+ij|2—11-