Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds?

Jiaxian WUQihua RUANYihu YANG

1 Introduction

The notion of Bakry-Emery Riccitensor associated with a dif f usion operator was introduced by Bakry[1],which we recall as follows.

Definition 1.1Given an n-dimensional Riemannian manifold(M,g)and a C2function φ on M,one has a dif f usion operator L:=+?φ·?,where theand?are the Laplace operator and the gradient operator on M respectively.Then the Bakry-Emery Riccitensor associated with the dif f usion operator L is defined as the following symmetric2-tensor:

where the constant m≥n;if m=n,we assume φ=0.Denote byRic∞the limit=Ric??2φ.

In this note,we want to study the gradient estimates of the Hamilton type for the positive solution to the following nonlinear dif f usion equation:

on a complete noncompact Riemannian manifold with the above Bakry-Emery Ricci curvature bounded below by?K(K≥0),wherea(x)andb(x)areC1functions with certain conditions(for details,see Theorem 1.3).

The elliptic case of the equation(1.1)withφ=0,namely

was first considered by Ma[6]in the case thataandbare constants anda<0 when he studied the gradient Ricci Soliton.He also pointed out that it is interesting to consider the gradient estimates for the positive solutions to the corresponding parabolic equation

Later,Yang[12]studied the above parabolic equation and obtained the gradient estimate of Li-Yau type(see[3])for the solutions to(1.3).Here we should also mention that Li[5]studied earlier the following equation:

for someα>0,and got the gradient estimates and the Harnack inequality which generalize the corresponding estimates of Li-Yau[3].

Whena=b=0,the equation was studied by Li[5].He obtained a gradient estimate of the Li-Yau type.

There is another kind of gradient estimates developed by Hamilton[2].He considered the heat equation on compact manifolds and obtained the following estimate,which we call the gradient estimate of Hamilton type.

Theorem 1.1Let M be a compact manifold without boundary and with Ricci curvature bounded below by?K,K≥0.Suppose that u is any positive solution to the heat equation ut=u with u≤C for all(x,t)∈M×(0,+∞).Then

In[10],Souplet and Zhang extend the above gradient estimate to noncompact manifolds.

Theorem 1.2(Souplet-Zhang)Let M be an n-dimensional complete noncompact manifold with the Ricci curvature bounded below by?K,K≥0.Suppose that u is any positive solution to the heat equation ut=u in Q2R,2T≡B(x0,2R)×[t0?2T,t0],and u≤C in Q2R,2T.Then one has in QR,T,

where C1is some positive constant depending only on the dimension n of M.

In this note,we will study the equation(1.1),and hope to get the gradient estimates of Hamilton type for the solutions under the condition of Bakry-Emery Ricci curvature.Our method is motivated by[9],where the second author obtained the gradient estimate of Hamilton type for the solution to the equation(1.1)whena(x)≡0.We should point out that the equation(1.1)is a nonlinear PDE whena(x)0.Our main result can be stated as follows.

Theorem 1.3Let(M,g)be an n-dimensional complete non-compact Riemannian manifold with Bakry-Emery curvature?K for some constant K≥0,p∈M,B(p,R)the geodesic ball with radius R and the center at p.Suppose that u(x,t)is a positive smooth solution to the di ff usion equation(1.1)in Q2R,2T0≡B(p,2R)×[0,2T0]?M×[0,∞).Let α=lnu+1,γ=max{1,|α?1|}and assume that≤C0,≤C0,|a|≤C0,|b|≤C0for some constant C0>0.Then one has the following estimate in QR,T0:

where C1is a general constant independent of the dimension n of M and depending only on C0.

SettingR→ +∞,we can get a global gradient estimate for the equation(1.1),which is independent of the dimensionnofM.

Corollary 1.1Under the conditions of Theorem1.3,then one has in M×[0,T0],

2 Gradient Estimates of Hamilton Type

In this section,we use a certain cut-of ffunction and the maximum principle to show Theorem 1.3.

We first state a general Laplacian comparison theorem(see[5,8]),which will be used in the following proof.Given ann-dimensional Riemannian manifold(M,g)and aC2functionφonM,one has a dif f usion operatorL:=+?φ·?,where theand?are the Laplace operator and the gradient operator onMrespectively.One has the so-called Bakry-Emery Riccitensor associated with the dif f usion operatorL.The comparison theorem associated withLcan be stated as follows.

Lemma 2.1Let M be an n-dimensional complete Riemannian manifold with the Bakry-Emery curvatureassociated with L greater than?K(K>0).One then has

where the Cpdenotes the cut locus of p.

We also need aC2cut-of ffunctionη=η(t)on[0,+∞)(also see[3]),which is defined as follows:

and satis fies that?t>0,0≥≥?C,(t)≥?C,whereCis a positive constant.Here and henceforth,unless otherwise stated,byC,C0,C1,etc,we always mean some general constants independent of the dimensionnofM.

Letρ(x)be the distance function atp,and de fine

Then we have

By(2.1),one then has

For convenience,we introduce two bilinear operators Γ and Γ2(see[1])as follows.Foru,v∈C2(M),set

and

A simple computation shows that?u∈C2(M),

and

where:=L??t.

In the following,we denote Γ(u,u)and Γ2(u,u)by Γ(u)and Γ2(u)respectively.Thus,by the assumption of the Bakry-Emory curvatureand the Bochner’s formula,one has,for anyu∈C2(M),

Proof of Theorem 1.3We now begin to show our main theorem.Setf=lnuand

Rewriting the equation(1.1)as

one then has

Thus,using(2.5)–(2.7),one has

We now estimate the last three terms in(2.8).

and

Substituting(2.9)–(2.11)into(2.8),one obtains

Let?=tψand suppose that the function?ωtakes the maximum at the point(x1,t1)inB(p,2R)×[0,T]?M×[0,T0].It is well-known(see[3])that one may assume thatx1is not in the cut locus ofp.Then the maximum principle implies

Thus,one has at(x1,t1)

Combining(2.12)with(2.14),one has at(x1,t1),

We now estimate the seventh,eighth and ninth items in the left side of(2.15).By the Young’s inequality,(2.4)and(2.13),we obtain at(x1,t1),

and

Substituting(2.16)–(2.18)into(2.15)at(x1,t1),one then has

Multiplying both sides of(2.19)byand noting that 0≤ψ≤1,one obtains at(x1,t1),

Setγ=max{1,|α?1|}.We observe that iff≤0,≤1 and iff≥0,≤(α?1)2.Using these together with(2.3)and the assumptions onaandb,(2.20)then becomes at(x1,t1)

whereC0,Care general constants independent of the dimensionnofM.

Sinceψ=1 onB(p,R)and(x1,t1)inB(p,2R)×[0,T]is the maximum point of?ω,then one has

?x∈B(p,R).

AsTis arbitrary,ω=|?ln(α?f)|2andf=lnu,one has at(x,t)∈B(p,R)×[0,T0],

This completes the proof of Theorem 1.3.

3 Some Further Remarks

From the proof of Theorem 1.3,we see that the curvature condition of≥?Kis only used in the proof of the inequality(2.6)and the Laplacian comparison theorem of the distance function(see Lemma 2.1).On the other hand,again by the Bochner formula,one still has

So if the curvature condition of≥ ?Kis replaced by Ric∞≥ ?K,the same inequality as(2.6)still works.However,the Laplacian comparison theorem of the distance function(see Lemma 2.1)does not work anymore under the curvature condition of Ric∞≥ ?K.

Fortunately,following Wei and Wylie’s idea(see[11]),we can get that if Ric∞≥ ?Kand|φ|≤k,then the similar Laplacian comparison theorem of the distance function still holds for the dif f usion operatorL= Δ+?φ·?:Lρ(x)≤Thus,by using the same argument as above,we have the following global gradient estimate for the equation(1.1).

Theorem 3.1Let(M,g)be an any-dimensional complete non-compact Riemannian manifold withRic∞≥?K and|φ|≤k for some positive constants K and k.Suppose that u(x,t)is a positive smooth solution to the di ff usion equation(1.1)on M×[0,T].Let α=lnu+1and assume that≤C0,≤C0,|a|≤C0,|b|≤C0for some positive constant C0.Then one has in M×[0,T],

where γ:=max{1,|α? 1|}and C1is a positive constant depending only on C0(in particular,independent of the dimension n of M).

If botha(x)andb(x)in the equation(1.1)are constant functions,then we have the following global gradient estimate.

Corollary 3.1Let(M,g)be an any-dimensional complete non-compact Riemannian manifold withRic∞≥ ?K and|φ|≤k for some positive constants K and k.Suppose that u(x,t)is a positive smooth solution on M×[0,T].Let α=lnu+1.Then one has in M×[0,T],

AcknowledgementThe author thanks the anonymous referee for his/her valuable comments and suggestions.

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