Harnack estimates for a nonlinear parabolic equation under Ricci flow

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Differential Geometry and its Applications

www.elsevier.com/locate/difgeo

Harnack estimates for a nonlinear parabolic equation under Ricci flow

Yi Li^a,∗, Xiaorui Zhu^b,c

a FacultyofScience,TechnologyandCommunication(FSTC),MathematicResearchUnit,Campus Belval,UniversiteduLuxembourg,MaisonduNombre,6,avenuedelaFonte,L-4364,Esch-sur-Alzette, Luxembourg

bCenterofMathematicalSciences,ZhejiangUniversity,Hangzhou,310027, China

cChinaMaritimePoliceAcademy,Ningbo,315801, China

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received7September2016 Availableonlinexxxx CommunicatedbyF.Fang

MSC:

53C44

Keywords:

Nonlinearparabolicequation Harnackestimate

Ricciflow

Inthispaper,weconsidertheHarnackestimatesforanonlinearparabolicequation undertheRicciflow.ThegradientestimatesforpositivesolutionsaswellasLi–Yau typeinequalitiesarealsogiven.

©2017ElsevierB.V.Allrightsreserved.

1. Introduction

Thenonlinearparabolicequationisaclassicalsubjectthathasbeen extensivelystudied,whichleadsto lotsofimportantresultsespeciallyinresearchesofdifferentialgeometry.Oneoftheimportanttechniquein studying theheatequation isthedifferential HarnackinequalitydevelopedbyLiandYau[7].This isalso appliedtoRicciflowbyHamilton[6],andplaysanimportantroleinsolvingthePoincaréconjecture[9].

NowwewillstudythecasewhereMisacompletemanifoldwithoutboundary.Considerpositivesolutions ofanonlinearparabolicequationonthemanifoldM,whichevolvesundertheRicciflow.A seriesofgradient estimatesareobtainedforsuchsolutions,includingseveralLi–Yau-typeinequalities.Let(M,g(t))_t∈[0,T] be acompletesolutiontotheRicciflow

✩ YiLi is partiallysupportedby theFondsNational de laRecherche Luxembourg (FNR)under the OPENscheme (project GEOMREVO14/7628746).XiaoruiZhuispartiallysupportedbyCPSF(grant)No.2014M551721andZhejiangProvinceNatural ScienceFoundationofChina(grant)No.Q14A010002.

* Correspondingauthor.

E-mailaddresses:[email protected](Y. Li),[email protected](X. Zhu).

https://doi.org/10.1016/j.difgeo.2017.10.017 0926-2245/©2017ElsevierB.V.Allrightsreserved.

∂_tg(t) =−2Ric_g(t), t∈[0, T]. (1.1) WeassumethatitsRiccicurvatureremainsuniformlyboundedforallt∈[0,T].Considerapositivefunction u=u(x,t) definedonM×[0,T] solvingtheequation

Δ_g(t)−q−∂t

u=au(lnu)^α, t∈[0, T], (1.2)

which hasbeen firststudiedin[14]where g(t)≡g is afixedmetric.Qian[10] and Wu[13] gotaseries of similar conclusions.HereΔ_g(t) standsfortheLaplacianof g(x,t) definedonM×[0,T] and q(x,t) isaC² functiondefinedonM×[0,T].NoticethattheLaplacianΔ_g(t)dependsontheparametert,andweshould study thenonlinearparabolic equation (1.2)coupledwiththeRicciflow (1.1).Theformula (1.1)provides us withadditionalinformationaboutthecoefficientsoftheoperatorΔ appearingin(1.2)butisitselffully independent of(1.2).

2. Gradientestimates I:α= 1

Firstly, we introduce a cut-off function (see [1,3,7,8,14]) on Bρ,T := {(χ,t) ∈ M × [0,T] : dist_g(t)(χ,x0) < ρ}, where dist_g(t)(χ,x0) stands for the distance between χ and x0 with respect to the metricg(t),whichsatisfiesa basicanalyticalresultstatedinthefollowing lemma.

Lemma2.1. Givenτ ∈(0,T],thereexistsasmoothfunctionΨ : [0,∞)×[0,T]→Rsatisfying thefollowing requirements:

(1) ThesupportofΨ(r,t)isasubsetof [0,ρ]×[0,T],0≤Ψ(r,t)≤1in[0,ρ]×[0,T],andΨ(r,t)= 1holds in[0,^ρ₂]×[τ,T].

(2) Ψisdecreasing asaradialfunctionin thespatialvariables.

(3) The estimate |∂tΨ|≤ ^C_τΨ^1/2 issatisfiedon[0,∞)×[0,T] forsomeC >0.

(4) The inequalities−^C_ρ^αΨ^α≤∂rΨ≤0and|∂_r²Ψ|≤^C_ρ^2αΨ^α holdon[0,∞)×[0,T]forevery a∈(0,1) with someconstant Cα dependenton a.

Proof. See[1]. 2

ThesepropertiesarederivedfromCalabi’sargument(see,e.g.,[2,4,11]).Usingthisauxiliaryfunctionand applyingthemaximumprinciple,weareabletoestablishLi–Yau-typeinequalityforthesystem(1.1)–(1.2).

Theorem2.2.Supposethat(M,g(t))_t∈[0,T]isacompletesolutiontotheRicciflow(1.1)onann-dimensional manifoldMwithsup_Bρ,T |Ric_g(t)|g(t)≤KforsomeK >0,anduisasmoothpositivefunctiononM×[0,T] satisfyingthenonlinearparabolicequation (1.2)wherethefunctionq(x,t)isdefined onM×[0,T]whichis C² in thex-variable andC¹ inthet-variable.Ifu(x,t)≤1on Bρ,T,α= 1,|∇g(t)q|g(t)≤γ,and

b:= 1

8+ min

M×[0,T]q−max{a,0}>0, then thereexistsaconstant C that dependsonly onnsuchthat

|∇g(t)u|g(t)

u ≤C

b 1

ρ+ 1

√t +√

K+ 1 +√γ+

|a| 1 + ln1 u

(2.1) on B_ρ/2,T with t= 0.

Whenqisnonnegativeanda≤0,theconstantC/bcanbeauniversalconstantwhichmeansaconstant dependingonlyonthedimensionn.Thenumber1/8 inbisnotessential,becauseinthefollowingproofwe shallseethatwecanreplace1/8 by 1.Toprovetheabovetheorem,weneedthefollowingcrucial lemma.

Lemma2.3.Let(M,g(t))_t∈[0,T] beacomplete solutiontotheRicciflow (1.1)onann-dimensionalmanifold Mwithsup_Bρ,T |Ric_g(t)|g(t)≤KforsomeK >0,anduisasmoothpositivefunctiononM×[0,T]satisfying thenonlinearparabolic equation (1.2) with α= 1,a<0.Weassume that u≤1 onB_ρ,T.If f := lnu and w:=|∇g(t)ln(1−f)|²g(t)=|∇g(t)f|²g(t)/(1−f)²,thentheinequality

(Δ−∂_t)w≥ 2f

1−f∇f,∇w+ 2(1−f)w² +2∇f,∇q

(1−f)² + 2a |∇f|²

(1−f)² +2|∇f|²(q+af)

(1−f)³ (2.2)

holdson Bρ,T.

Proof. Sinceuisapositivesolutionto thenonlinearparabolicequation (1.2)withα= 1 anda<0,direct calculationshowsthat

Δf +|∇f|²−ft−q−af = 0, ft:=∂tf.

Thepartialderivativeofwwithrespect tot isgivenby wt= 2∇f,∇ft

(1−f)² +2|∇f|²ft

(1−f)³ +2Ric(∇f,∇f) (1−f)²

= 2∇f,∇(Δf +|∇f|²−q−af)

(1−f)² +2Ric(∇f,∇f) (1−f)² + 2|∇f|²(Δf+|∇f|²−q−af)

(1−f)³ .

UsingBochner’sidentity ∇f,∇Δf=∇f,Δ∇f−Ric(∇f,∇f) weobtain w_t= 2∇f,Δ∇f+ 2∇f,∇(|∇f|²−q−af)

(1−f)² +2|∇f|²(Δf+|∇f|²−q−af) (1−f)³ . Thepartialderivativeofwwithrespect toxisgivenby

Δw= 2 |∇²f|²

(1−f)² + 2∇f,Δ∇f

(1−f)² + 4∇f,∇|∇f|²

(1−f)³ + 6 |∇f|⁴

(1−f)⁴ + 2|∇f|²Δ_g(t)f (1−f)³ . Combiningthose partialderivativesimply

(Δ−∂_t)w= 2 |∇²f|²

(1−f)² + 4∇f,∇|∇f|²

(1−f)³ + 6 |∇f|⁴

(1−f)⁴ −2∇f,∇|∇f|² (1−f)²

−2 |∇f|⁴

(1−f)³ + 2a |∇f|²

(1−f)² + 2∇f,∇q

(1−f)² + 2 q|∇f|²

(1−f)³+ 2a f|∇f|² (1−f)³. Ontheotherhand,thegradientterm∇f,∇wisdeterminedby

∇f,∇w=∇f,∇|∇f|²g(t)

(1−f)² + 2 |∇f|⁴ (1−f)³.

Pluggingitinto theevolutionof (Δ−∂_t)wweconcludethat (Δ−∂t)w= 2 |∇²f|²

(1−f)² + 2∇f,∇|∇f|² (1−f)³ + 2

1−f∇f,∇w+ 2 |∇f|⁴ (1−f)⁴

−2∇f,∇w+ 2 |∇f|⁴

(1−f)³ + 2a |∇f|² (1−f)² + 2∇f,∇q

(1−f)² + 2 q|∇f|²

(1−f)³+ 2a f|∇f|² (1−f)³. Because oftheidentity

|∇²f|²

(1−f)² +∇f,∇|∇f|²

(1−f)³ + |∇f|⁴ (1−f)⁴ =

∇²f

1−f +∇f⊗ ∇f (1−f)²

²

we thereforearriveat (Δ−∂_t)w= 2

∇²f

1−f +∇f⊗ ∇f (1−f)²

²+ 2

1−f∇f,∇w+ 2 |∇f|⁴

(1−f)³ −2∇f,∇w + 2∇f,∇q

(1−f)² + 2a |∇f|²

(1−f)² +2|∇f|²(q+af) (1−f)³ whichimmediately implies

(Δ−∂t)w≥ 2f

1−f∇f,∇w+ 2|∇f|⁴

(1−f)³ +2∇f,∇q

(1−f)² +2a|∇f|²

(1−f)² +2|∇f|²(q+af) (1−f)³ . This completetheproof. 2

Proof of Theorem 2.2. Pick a number τ ∈ (0,T] and fix a function Ψ(x,t) satisfying the conditions of Lemma 2.1.Wewillestablish(2.1)at(x,τ) forallxsuchthatdist_g(τ)(x,x0)< ^ρ₂.DefineΨ:M×[0,T]→R bysetting

Ψ(x, t) := Ψ

dist_g(t)(x, x0), t .

Then, using(2.2)andastraightforwardcalculation,onehas (Δ−∂t) (Ψw)≥Ψ−Λ,∇w+

2(1−f)w²+2∇f,∇q

(1−f)² + 2a |∇f|² (1−f)² + 2|∇f|²(q+af)

(1−f)³ Ψ + 2∇w,∇Ψ+wΔΨ−w∂tΨ

≥ −Λ,∇(Ψw)+ 2Ψ(1−f)w²+wΛ,∇Ψ +wΔΨ−wΨ_t+ 2

Ψ∇Ψ,∇(Ψw) −2|∇Ψ|²

Ψ w

+

2∇f,∇q

(1−f)² + 2a |∇f|²

(1−f)² +2|∇f|²(q+af) (1−f)³ Ψ

where Λ=−_1−f^2f ∇f. Byourassumptionthat|Ric|≤K onB_ρ,T andLemma 2.1that−^C_ρ¹Ψ^1/2≤Ψ_r≤0, and theidentity

−wΨ_t=−

Ψ_t+ Ψ_r∂_tdist_g(t)(·, x₀) w, wehave(because−∂_tdist_g(t)(·,x₀)≤4

(m−1)K,cf. Lemma 8.33in[5])

−wΨ_t≥ −Ψ_tw−4C1

(m−1)K ρ wΨ^1/2.

SupposethatΨw achievesitsmaximumat(x0,t0).By[7],withoutloss ofgenerality,wemayassumethat x0isnotinthecut-locusofM.Atthepoint(x0,t0),onehasΔ(Ψw)≤0,∇(Ψw)= 0,(Ψw)t≥0.Therefore

2Ψ(1−f)w²≤ −wΛ,∇Ψ+ 2|∇Ψ|²

Ψ w−wΔΨ +wΨ_t

−

2∇f,∇q

(1−f)² + 2a |∇f|²

(1−f)²+2|∇f|²(q+af)

(1−f)³ Ψ (2.3)

at(x₀,t₀).Weneedtobound eachtermontheright-handsideof(2.3):

|wΛ,∇Ψ| ≤2w|f|

1−f|∇f||∇Ψ|= 2w^3/2|f||∇Ψ| ≤Ψ(1−f)w²+27 16

|f|⁴|∇Ψ|⁴ [Ψ(1−f)]³

where we used theYoung’s inequality thatab ≤a^p+b^q/(q(p)^q/p) for any a,b,> 0 andp,q > 1 with p⁻¹+q⁻¹= 1.Thistogetherwith Lemma 2.1implies

|wΛ,∇Ψ| ≤Ψ(1−f)w²+C2

f⁴

ρ⁴(1−f)³. (2.4)

UsingagainLemma 2.1wehave

|∇Ψ|²

Ψ w= Ψ^1/2w|∇Ψ|² Ψ^3/2 ≤ 1

8Ψw²+ 2|∇Ψ|⁴ Ψ³ ≤1

8Ψw²+C3

ρ⁴. (2.5)

Furthermore,bythepropertiesofΨ andtheassumption oftheRiccicurvature, onehas(cf.,[12,15])

−wΔΨ≤ 1

8Ψw²+C4

ρ⁴ +C4K². (2.6)

TheestimationforwΨ_tisgiven by(cf. [1])

|wΨt| ≤1

8Ψw²+C5

τ² +C5K². (2.7)

Sincef ≤0 it followsthat

2∇f,∇q (1−f)² Ψ

≤ |∇f|²+|∇q|²

(1−f)² Ψ ≤ wΨ +γ²Ψ

≤ 1

8Ψw²+ (2 +γ²)Ψ, (2.8)

2a |∇f|² (1−f)²Ψ

= 2awΨ ≤ 1

8Ψw²+ 2a²Ψ (2.9)

and

−2|∇f|²(q+af)

(1−f)³ Ψ = 2Ψw² −q

1−f +a −f 1−f

≤2Ψw²

− min

M×[0,T]q+a −f 1−f

(2.10)

≤2

max{a,0} − min

M×[0,T]q

Ψw². Substituting(2.5)–(2.10)totheright-handsideof(2.3),wededucethat

Ψ(1−f)w²≤C6

f⁴

ρ⁴(1−f)³ +3

4Ψw²+ 2

max{a,0} − min

M×[0,T]q

Ψw² + C6

τ² +C6

ρ⁴ +C6K²+

2 +γ²+ 2a² at (x₀,t₀).Sincef <0,thenf⁴/(1−f)⁴≤1.Itfollowsthat

1

4+ 2 min

M×[0,T]q−2 max{a,0}

Ψw²≤ 2C6

ρ⁴ +C6

τ² +C₆K²+

2 +γ²+ 2a² at (x0,t0).When

1

8+ min

M×[0,T]q−max{a,0}>0 (2.11) we canconcludethat

Ψw²≤ C7

1

8+ min_M×[0,T]q−max{a,0} 1

ρ⁴ + 1

τ² +K²+ 1 +γ²+a²

at (x₀,t₀).BecauseΨ(x,τ)= 1 whendist_g(τ)(x,x₀)< ρ/2,wefinally arriveat w²(x, τ)≤ C7

1

8+ min_M×[0,T]q−max{a,0} 1

ρ⁴ + 1

τ² +K²+ 1 +γ²+a²

onB_ρ/2,T,which,sinceτ∈(0,T] wasarbitrary,implies

|∇f|

1−f ≤ C8

1

8+ min_M×[0,T]q−max{a,0} 1

ρ+ 1

√t+√

K+ 1 +√ γ+

|a|

.

Wehavecompletedtheproof ofTheorem 2.2.

Thenumber1/8 in(2.11)isnotessential,becauseintheaboveargumentwecanreplace1/8 in(2.5)–(2.9) byagivenpositivenumber,and henceweneedonlyto requirethat

1−5+ 2 min

M×[0,T]q−2 max{a,0}>0 insteadof(2.11).When

1 + 2 min

M×[0,T]q−2 max{a,0} is positive,wechoosetobe thisnumberdividedby5.

3. GradientestimatesII:generalcase

InthissectionweextendTheorem 2.2with α= 1 tothegeneralcase.

Lemma 3.1. Suppose (M,g(t))t∈[0,T] is a complete solution to the Ricci flow (1.1) on an n-dimensional manifold M, with −K1g(t) ≤Ric_g(t) ≤K2g(t) for some K1,K2 > 0 on Bρ,T. If u is a smooth positive function satisfying the nonlinear parabolic equation (1.2), then, for given β ≥ 1 and any c,d > 0 with c+d= 1/β,wehave

(Δ−∂t)F≥ −2∇f,∇F −F

t +2cβt n

|∇f|²−q−ft−af^α2

−2(β−1)t∇f,∇q −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−2βtK1|∇f|²−nβt

2d K² (3.1)

−βaαtf^α−1

−|∇f|²+ft+q+af^α

−βtΔq,

whereK:= max{K1,K2},f := lnu,andF :=t(|∇f|²−βft−βq−βaf^α).

Proof. Theproofofthislemmawasoriginal from[1].Nowwewillfindaconvenientbound onΔF like the wayin[16].Notice that

∇iF =t

2∇^jf∇i∇jf−β∇ift−β∇iq−βaαf^α−1∇if .

ThentheLaplaceofF equals

ΔF =∇ⁱ∇iF

=t

2|∇²f|²+ 2∇f,Δ∇f −βΔft−βΔq

−βaα

(α−1)f^α−2|∇f|²+f^α−1Δf .

UsingtheBochner’sformulaΔ∇f =∇Δf+ Ric(∇f,·),weget ΔF=t

2|∇²f|²+ 2∇f,∇Δf+ 2Ric(∇f,∇f)−βΔft

−βΔq−βaα

(α−1)f^α−2|∇f|²+f^α−1Δf

≥t

2(Δf)²

n + 2∇f,∇Δf −2K1|∇f|²−β(Δf)t−βΔq

−βaα

(α−1)f^α−2|∇f|²+f^α−1Δf

+ 2βR_ij∇ⁱ∇^jf since|∇²f|²≥ _n¹(Δf)² andΔft= (Δf)t−2Rij∇ⁱ∇^jf.Recallingfromtheresult

Δf =−|∇f|²+q+f_t+af^α=−F

t −(β−1) (q+f_t+af^α), wearriveat

ΔF ≥ 2cβt n

|∇f|²−q−ft−af^α2

+ 2dβt

n (Δf)²+ 2tβRij∇ⁱ∇^jf

−2t

∇f,∇ F

t + (β−1)(q+ft+af^α)

−2K1t|∇f|²−tβ

−F

t −(β−1)(q+ft+af^α)

t

−βtΔq (3.2)

−βaαt

(α−1)f^α−2|∇f|²+f^α−1Δf inthesetBρ,T.Because

2dβt

n (Δf)²+ 2tβRij∇ⁱ∇^jf =2dβt n

(Δf)²+n

dRij∇ⁱ∇^jf

=2dβt n

∇²f+ n

2dRic²−nβt 2d |Ric|²

≥ −nβt 2d|Ric|², theinequality(3.2)canbe writtenas

ΔF ≥ 2cβt n

|∇f|²−q−f_t−af^α2

−nβt 2d|Ric|²

−2t

∇f,∇ F

t + (β−1)(q+f_t+af^α)

−2K₁t|∇f|²−tβ

−F

t −(β−1)(q+f_t+af^α)

t

−βtΔq (3.3)

−βaαt

(α−1)f^α−2|∇f|²+f^α−1Δf .

To getthetime derivativeofF,weshallusetheidentity Ft= F

t +t

|∇f|²−βft−βq−aβf^α

t

.

Subtractingthisfrom (3.3),weget

(Δ−∂t)F≥ −2∇f,∇F −F

t +2cβt n

|∇f|²−q−ft−af^α2

−βtΔq−2(β−1)t∇f,∇q −nβt 2d |Ric|²

−2(β−1)taαf^α−1|∇f|²−βtaα(α−1)f^α−2|∇f|²

−βaαtf^α−1

− |∇f|²+ft+q+af^α

−2βK1t|∇f|².

Now theinequality(3.1)followsimmediately bynotingthat|Ric_g(t)|²_g(t)≤K². 2

Nowwecanconsiderthelocalspace–timegradientestimatewithLemma 2.3.Inthefollowingpart,nis thedimensionofM.

Theorem3.2.Supposethat(M,g(t))_t∈[0,T]isacompletesolutiontotheRicciflow(1.1)onann-dimensional manifold M with −K₁g(t) ≤ Ric_g(t) ≤ K₂g(t) for some K₁,K₂ > 0 on B_ρ,T. If u is a smooth positive

functionsatisfyingthenonlinearparabolicequation (1.2),thenthereexistsaconstantC depending onlyon nsuch that,onB_ρ/2,T,

(1) fora≥0,wehave

|∇g(t)f|²g(t)−βft−βq−βaf^α≤ nβ

2c(1−)t+(A+γ)nβ

2c(1−) + n²β³C² 4c²(1−)(β−1)ρ² + nβ[βK1+a(β−1)α|f^α−1|∞]

c(1−)(β−1) + nβ²aα|α−1||f^α−2|∞

2c(β−1)(1−) +

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) ,

(2) fora≤0,wehave

|∇g(t)f|²g(t)−βf_t−βq−βaf^α≤ nβ

2c(1−)t+(A+γ)nβ

2(1−c) + n²β³C₁² 4c²(1−)(β−1)ρ² + nβ[βK1−^a₂(β−1)α|f^α−1|∞]

c(1−)(β−1) +

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) .

Heref := lnu,|f|∞:= maxM|f|,K:= max{K1,K2},β >1,0< <1,|∇g(t)q|g(t)≤γ,Δ_g(t)q≤θ, A=C

1 ρ² +

√K₁ ρ +1

t +K

andc,d>0with c+d= 1/β.

Proof. Wewill usethesamenotation f = lnuandF =t(|∇f|²−βf_t−βq−βaf^α) as inLemma 3.1.For thefixedτ ∈(0,T],chosethecut-offfunctionΨ constructedinLemma 2.1. DefineΨ:M×[0,T]→R by setting

Ψ(x, t) = Ψ

dist_g(t)(x, x₀), t .

Lemma 3.1impliesthat

(Δ−∂t) (ΨF)≥ −2∇f,∇(ΨF)+ 2F∇f,∇Ψ +

2cβt n

|∇f|²−q−f_t−af^α2

−F

t − βtΔq

−2(β−1)t∇f,∇q −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−2βtK1|∇f|²−nβt 2d K²

−βaαtf^α−1

−|∇f|²+ft+q+af^α Ψ +FΔΨ + 2

Ψ∇Ψ,∇(ΨF) −2|∇Ψ|²

Ψ F−F∂Ψ

∂t.

Let(x₀,t₀) beamaximumpointforthefunctionΨF intheset{(x,t)|0≤t≤τ,d_g(t)(x,x₀)≤ρ}.Thenat thepoint (x₀,t₀) wehave

0≥2F∇f,∇Ψ+F(Δ−∂t) Ψ−2|∇Ψ|²

Ψ F

+

−F

t +2cβt n

|∇f|²−q−ft−af^α2

−βtΔq

−2(β−1)t∇f,∇q −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−βaαtf^α−1

−|∇f|²+ft+q+af^α

−2βtK1|∇f|²−nβt 2dK² Ψ.

ByLemma 2.1andtheLaplaciancomparison theorem,wehave

|∇Ψ|²

Ψ ≤C_1/2² ρ² , ΔΨ≥ −C_1/2Ψ^1/2

ρ² −C_1/2Ψ^1/2

ρ (n−1)

K1coth(

k1ρ)

≥ −d₁

ρ² −d₁Ψ^1/2 ρ

K1,

−∂_tΨ≥ −CΨ^1/2

τ −C_1/2KΨ^1/2

where C_1/2,C and d₁ are positive constants depending only on n. Plugging those estimates into above inequalityyieldsthat

0≥d2

−1

ρ² −Ψ^1/2 ρ

K1−Ψ^1/2

τ −KΨ^1/2

F−2F|∇f||∇Ψ| +

2cβt n

|∇f|²−q−ft−af^α2

−F

t − βtΔq−nβt 2d K²

−2(β−1)t∇f,∇q −2(β−1)taαf^α−1|∇f|²−2βtk1|∇f|² (3.4)

−βtaα(α−1)f^α−2|∇f|²−βaαtf^α−1

−|∇f|²+f_t+q+af^α Ψ at (x₀,t₀),whered₂ isequalto max{d₁,C,C_1/2,2C_1/2² }.Introduceafunction

A:=d₂ 1

ρ²+Ψ^1/2 ρ

K₁+Ψ^1/2

τ +KΨ^1/2

.

If onemultipliesbytΨ andmakesafewelementarymanipulations,onewillobtain 0≥ −2F|∇f||∇Ψ|Ψt−AFΨt+

2cβt n

|∇f|²−q−f_t−af^α2

−βtΔq−2(β−1)t∇f,∇q −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−βaαtf^α−1

−|∇f|²+f_t+q+af^α

(3.5)

−2βtK₁|∇f|²−nβt

2dK² Ψ²t−FΨ²

at(x₀,t₀).Asin[3,14],weset

μ:=|∇f|²(x0, t0) F(x₀, t₀) ≥0.

Because|∇f|=μ^1/2F^1/2 and

|∇f|²−f_t−q−af^α=F

μ−μt−1 βt

,

∇f,∇Ψ ≤ |∇f||∇Ψ| ≤ C1

ρ Ψ^1/2|∇f| wecansimplify(3.5)into thefollowing inequality

AF tΨ≥ −2C₁t

ρ Ψ^3/2μ^1/2F^3/2−Ψ²F+2cΨ²

nβ [1 + (β−1)μt]²F²−nβ

2dK²(Ψt)²

−2(tΨ)²[βK₁+a(β−1)αf^α−1]μF+atΨ²αf^α−1[1 + (β−1)tμ]F (3.6)

−β(tΨ)²θ−2(β−1)(tΨ)²γ(μF)^1/2−β(tΨ)²aα(α−1)f^α−2μF at(x₀,t₀).Ifweset G:= ΨF,thenat thepoint(x₀,t₀) theinequality(3.6)becomes

AtG≥ −2C₁t

ρ μ^1/2G^3/2−ΨG+ 2c

nβ[1 + (β−1)μt]²G²−nβ

2dK²(Ψt)²

−2Ψt²[βK₁+a(β−1)αf^α−1]μG+aΨtαf^α−1[1 + (β−1)μt]G (3.7)

−β(Ψt)²θ−2(β−1)t²Ψ^3/2γμ^1/2G^1/2−βt²Ψaα(α−1)f^α−2μG at(x₀,t₀).Accordingto theCauchyinequality,where0< <1,

2C₁t

R μ^1/2G^3/2≤ 2c

nβ[1 + (β−1)μt]²G²+ nβC₁²t²μG 2cρ²[1 + (β−1)μt]², 2μ^1/2G^1/2≤1 +μG,

wecansimplify(3.7)as

AtG≥2c(1−)

nβ [1 + (β−1)μt]²G²−ΨG− nβ²C₁²t²μ 2cρ²[1 + (β−1)μt]²G

−2Ψt²[βK1+a(β−1)αf^α−1]μG+aΨtαf^α−1[1 + (β−1)μt]G

−βΨ²t²θ−(β−1)t²Ψ^3/2γ−(β−1)t²Ψ^3/2γμG

−βt²Ψaα(α−1)f^α−2μG−nβ

2dK²(Ψt)², at(x0,t0),or equivalently,

2c(1−)[1 + (β−1)μt]²G²

nβ ≤

nβ²C₁²t²μ

2cρ²[1 + (β−1)μt]² +βt²Ψaα(α−1)f^α−2μ

−aΨtαf^α−1[1 + (β−1)μt] + (β−1)t²Ψ^3/2γμ

+ 2Ψt²[βK1+a(β−1)αf^α−1]μ+At+ Ψ G +

βΨ²θ+ (β−1)Ψ^3/2γ+nβ

2dK²Ψ² t² at (x₀,t₀).Notethat0≤Ψ≤1 and1+ (β−1)μt₀≥1.Therefore

2c(1−)G²

nβ ≤

At+ 1 + nβ²C₁²t²μ

2cρ²[1 + (β−1)μt]+2Ψt²[βK1+a(β−1)αf^α−1]μ [1 + (β−1)μt]²

− aΨtαf^α−1

1 + (β−1)μt + (β−1)γt²μ

1 + (β−1)μt +βt²Ψaα|α−1|f^α−2μ 1 + (β−1)μt G +

βθ+ (β−1)γ+nβ 2dK² t²

≤

At+ 1 + nβ²C₁²t

2cρ²(β−1)+2Ψt²[βK1+a(β−1)α|f^α−1|∞]μ

[1 + (β−1)μt]² (3.8)

+γt− aΨtαf^α−1

1 + (β−1)μt+βtΨaα|α−1||f^α−2|∞

β−1 G

+

βθ+ (β−1)γ+nβ 2dK² t².

Beforecompletingtheproof,werecallafact:ifx²≤ax+bforsomea,b,x≥0,then x≤a

2+

b+ a

2 2

≤a 2+√

b+a

2 =a+√

b. (3.9)

If a≥0 in(3.8),then from(3.8)wededucethat G²≤

(A+γ)nβt

2c(1−) + nβ

2c(1−)+ n²β³C₁²t 4c²(1−)ρ²(β−1) + nβ²aα|α−1||f^α−2|∞t

2c(β−1)(1−) +nβ[βK₁+a(β−1)α|f^α−1|∞]t

c(1−)(β−1) G (3.10)

+ [βθ+ (β−1)γ+^nβ_2dK²]nβt²

2c(1−) .

Applying (3.9)to theinequality(3.10), wegetanupperbound forG:

G≤

(A+γ)nβ

2c(1−) + n²β³C₁²

4c²(1−)(β−1)ρ² +nβ[βK1+a(β−1)α|f^α−1|∞] c(1−)(β−1) + nβ²aα|α−1||f^α−2|∞

2c(β−1)(1−) τ+

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) τ+ nβ

2c(1−),

since t₁ ≤τ. Bythe constructionof Ψ, we havesup_Bρ/2,TF ≤sup_Bρ,T(ΨF)≤G(x₀,t₀) forall t∈ [0,τ].

Because τ≤T isarbitrary,itfollowsthat

|∇f|²−βf_t−βq−βaf^α≤ nβ

2c(1−)t+(A+γ)nβ

2c(1−) + n²β³C₁² 4c²(1−)(β−1)ρ² + nβ[βK₁+a(β−1)α|f^α−1|∞]

c(1−)(β−1)

+ nβ²aα|α−1||f^α−2|∞

2c(β−1)(1−) +

[βθ+ (β−1)γ+^nβ_2dK²]nβ 2c(1−)

where|f|∞:= maxM|f|.Similarly,whena≤0,we have G²≤

(A+γ)nβt

2c(1−) + nβ

2c(1−)+ n²β³C₁²t

4c²(1−)ρ²(β−1)+ nβ²K₁t c(1−)(β−1)

− nβatα|f^α−1|∞

2c(1−) G+[βθ+ (β−1)γ+^nβ_2dK²]nβt²

2c(1−) . (3.11)

From(3.9),(3.11),andaboveargument,anupperboundfordesiredquantityinthiscaseis

|∇f|²−βft−βq−βaf^α≤ nβ

2c(1−)t +(A+γ)nβ

2c(1−) + n²β³C₁² 4c²(1−)(β−1)ρ² + nβ[βK1−^a₂(β−1)α|f^α−1|∞]

c(1−)(β−1) +

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) .

Hence,wecompletetheproof. 2 Acknowledgements

TheauthorsthankforProfessorKefeng Liu’sconstantguidanceandhelp.

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