Liang ZHAO Zongwei MA

1 Introduction

In this paper,we study the following nonlinear parabolic equation:

on complete noncompact manifoldsMwith Ricci curvature bounded from below,where 0<σ<1 is a real constant,andb(x,t)is a function which isC2in thex-variable andC1in thet-variable.

Gradient estimates play an important role in the study of the PDE,especially the Laplacian equation and the heat equation.Li and Yau[5]developed the fundamental gradient estimate,which is now widely called the Li-Yau estimate,for any positive solutionu(x,t)to the heat equation on a Riemannian manifoldM,and showed how the classical Harnack inequality can be derived from their gradient estimate.Later,Hamilton[3]got the matrix Harnack estimate for the heat equation.

Let(Mn,g)be ann-dimensional complete noncompact Riemannian manifold.For a smooth real-valued functionfonMn,the drifting Laplacian is defined by

There is a naturally associated measure dμ=e−fdVonMnwhich makes the operatorself-adjoint.TheN-Bakry-Emery Ricci tensor is defined by

forandN=0 if and only iff=0.Here∇2is the Hessian operator and Ric is the Ricci tensor.Huang and Li[4]considered the generalized equation

on Riemannian manifolds and got some interesting gradient estimates.Wu[6]gave a local Li-Yau-type gradient estimate for the positive solutions to a general nonlinear parabolic equation

inM×[0,τ],wherea∈R,φis aC2-smooth function andq=q(x,t)is a function which generalizes many previous well-known results about gradient estimates.Zhang and Ma[7]considered gradient estimates on positive solutions to the following nonlinear equation:

on complete noncompact manifolds,and the authors got a gradient estimate for positive solutions of the above equation(1.2)whenNis finite and theN-Bakry-Emery Ricci tensor is bounded from below.

Theorem 1.1(see[7])Let(Mn,g)be a complete noncompact n-dimensional Riemannian manifold with the N-Bakry-Emery Ricci tensor bounded from below by the constant−K=:−K(2R),where R>0and K(2R)>0in the metric ball Bp(2R)around p∈M.Let u be a positive solution to(1.2).Then

(1)if c>0,we have

(2)if c<0,we have

where

Recently,Zhu[8]investigated the nonlinear parabolic equation

where 0<α<1,andλ(x,t)is a function defined onwhich isC1in the first variable andC0in the second variable.The author got a Hamilton-type estimate and a Liouville-type theorem for positive solutions to(1.3).

Theorem 1.2(see[8])Let(Mn,g)be a Riemannian manifold of dimension n≥2withRic(M)≥ −k for some k≥0.Suppose that u is a positive solution to(1.3)inSuppose also that u≤M and in QR,T.Thenthere exists a constant C=C(α,M),such that

in

In this paper,we will study the interesting Li-Yau type estimate for the positive solutions to(1.1).

Motivated by the above work,we present our main results about(1.1)as follows.

Theorem 1.3Let(Mn,g)be a complete noncompact n-dimensional Riemannian manifold with Ricci curvature bounded from below by the constant−K=:−K(2R),where R>0and K(2R)>0in the metric ball Bp(2R)around p∈M.Assume that|b|≤λ(2R),?b≤θ(2R)and|∇b|≤γ(2R)in Bp(2R)×[0,T)for some constants λ(2R),θ(2R)and γ(2R).Let u be a positive solution to(1.1)with u≤M1.Then for any constant0<β<1,if β<σ<1,we have

where c1and c2are positive constants and?∈(0,1).

Let,and we can get the global Li-Yau type gradient estimates for the nonlinear parabolic equation(1.1).

Corollary 1.1Let(Mn,g)be a complete noncompact n-dimensional Riemannian manifold with Ricci curvature bounded from below by the constant−K,where K>0.Assume that|b|≤λ(Mn),?b≤θ(Mn)and|∇b|≤γ(Mn)in Mn×[0,T)for some constants λ,θ and γ.Let u be a positive solution to(1.1)with u≤M1.Then for any constant0<β<1,if β<σ<1,we have

where

c1and c2are positive constants and?∈(0,1).

As an application,we get the following Harnack inequality.

Theorem 1.4Let(Mn,g)be a complete noncompact n-dimensional Riemannian manifold withRic(Mn)≥−K,where K>0.Assume that b is a nonpositive constant andLet u(x,t)be a positive smooth solution to the equation

on Mn×[0,+∞).Then if β<σ<1,for any points(x1,t1)and(x2,t2)on Mn×[0,+∞)with0

2 Proof of Theorem 1.3

Letube a positive solution to(1.1).Setw=lnu.Thenwsatisfies the equation

Lemma 2.1Let(Mn,g)be a complete noncompact n-dimensional Riemannian manifold with Ricci curvature bounded from below by the constant−K=:−K(2R),where R>0and K>0in the metric ball Bp(2R)around p∈Mn.Let w be a positive solution to(2.1).Then

where

and β is any constant satisfying0<β<1.

ProofDefine

By the Bochner formula,we have

Noticing that

and

we know

By(2.2)–(2.3),we obtain

and

So,we have

and

This implies that

We complete the proof of Lemma 2.1.

Proof of Theorem 1.3We take aC2cut-off functiondefined on[0,∞),such thatforforr∈[2,∞),and.Furthermore,?ϕsatisfies

and

for some absolute constantsc1,c2>0.Denote byr(x)the distance betweenxandpinM.Set

Using an argument of Cheng and Yau[2],we can assumeϕ(x)∈C2(M)with support inBp(2R).Direct calculation shows that onBp(2R),

By the Laplacian comparison theorem in[1],

ForT≥0,let(x0,t0)be a point inB2R(p)×[0,t]at whichϕFattains its maximum valueP.We assume thatPis positive(otherwise the proof is trivial).At the point(x0,t0),we have

It follows that

This inequality together with the inequalities(2.4)–(2.5)yields

where

At(x0,t0),by Lemma 2.1 and(2.6),we have

Multiplying both sides of the above inequality byt0ϕ,and noting the fact that 0<ϕ<1,we have

Let

It follows that

Following the method in[5,pp.161–162],we know

for any 0

Therefore,in our case,we have

Noticing that

we obtain

where

and

From the inequalitywe have

We can get

Notice that for allt∈[0,T],

We complete the proof of Theorem 1.3.

Proof of Theorem 1.4For any points(x1,t1)and(x2,t2)onM×[0,+∞)with 0

which means that

Therefore,

AcknowledgementsThe authors would like to thank the editors and the anonymous referees for their valuable comments and helpful suggestions that helped to improve the quality of this paper.Moreover,the authors would like to thank their supervisor Professor Kefeng Liu for his constant encouragement and help.

[1]Aubin,T.,Nonlinear Analysis on Manifolds,Springer-Verlag,New York,1982.

[2]Cheng,S.Y.and Yau,S.T.,differential equations on Riemannian manifolds and their geometric applications,Commun.Pure Appl.Math.,28,1975,333–354.

[3]Hamilton,R.,A matrix Harnack estimate for the heat equation,Comm.Anal.Geom.,1,1993,113–126.

[4]Huang,G.Y.and Li,H.Z.,Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian,2012.arXiv:math.DG/1203.5482v1

[5]Li,P.and Yau,S.T.,On the parabolic kernel of the Schrodinger operator,Acta Math.,156,1986,153–201.

[6]Wu,J.Y.,Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds,J.Math.Anal.Appl.,369,2010,400–407.

[7]Zhang,J.and Ma,B.Q.,Gradient estimates for a nonlinear equation?fu+cu−α=0 on complete noncompact manifolds,Comm.Math.,19,2011,73–84.

[8]Zhu,X.B.,Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds,Nonlinear Anal.,74,2011,5141–5146.