GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQU

Xiaobao ZHU（朱晓宝）

Department of Mathematics，School of Information，Renmin University of China，Beijing 100872，China

E-mail∶zhuxiaobao@ruc.edu.cn



GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS∗

Xiaobao ZHU（朱晓宝）

Department of Mathematics，School of Information，Renmin University of China，Beijing 100872，China

E-mail∶zhuxiaobao@ruc.edu.cn

In this article，we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations）u（x，t）+q（x，t）u（x，t）=0 and nonlinear parabolic equations）u（x，t）+h（x，t）up（x，t）=0（p＞1）on Riemannian manifolds. As applications，we obtain some theorems of Liouville type for positive ancient solutions of such equations.Our results generalize that of Souplet-Zhang（［1］，Bull.London Math.Soc. 38（2006），1045-1053）and the author（［2］，Nonlinear Anal.74（2011），5141-5146）.

Gradient estimate；linear parabolic equation；nonlinear parabolic equation；Liouville type theorem

2010 MR Subject Classification35K55；35B53

1　Introduction

In this article，we will study the linear parabolic equation

and the nonlinear parabolic equation

on complete noncompact Riemannian manifolds.The functions q（x，t）and h（x，t）are assumed to be C1in the first variable and C0in the second variable.In the celebrated article［3］，Li-Yau derived gradient estimates for positive solutions of equation（1.1）on a general Riemannian manifold.Using the gradient estimates，they obtained Harnack inequalities；especially，they obtained upper and lower bounds for the heat kernel.This result was generalized by J.Y.Li［4］to the case of（1.2）with 0＜p＜In 1993，R.Hamilton［5］obtained a new elliptictype gradient estimate，which is called Hamilton's gradient estimate in literature，for positive solutions of the heat equation on a closed Riemannian manifold，namely

Theorem（Hamilton［5］）Let M be a closed Riemannian manifold of dimension n≥2 with Ricci（M）≥−k for some k≥0.Suppose that u is any positive solution to the heat equation with u≤A for all（x，t）∈M×（0，∞）.Then，we have

It follows from Hamilton's gradient estimate that the temperature of two different points can be compared when the temperature is bounded.We remind the reader that Li-Yau's result［3］only allows us to compare the temperature of two different points at different times.

In［1］，Souplet and Zhang generalized Hamilton's estimate to complete noncompact Riemannian manifold case，and they derived a local elliptic gradient estimate for positive solutions of the heat equation and obtained Liouville theorem for positive ancient solutions of the heat equation.In［6］，B.Kotschwar used Hamilton's estimate to obtain a global gradient estimate for heat kernel on complete noncompact manifold.In［7］，M.Bailesteanu，X.Cao，and A.Pulemotov generalized Souplet-Zhang's result to Ricci flow case.For the nonlinear diffusion equation ut=∆u−∇φ·∇u−aulogu−bu，J.Wu［8］obtained a localized elliptic type gradient estimate for positive solutions；the Li-Yau type gradient estimate for this equation was obtained by Y. Y.Yang［9］and L.Chen and W.Y.Chen［10］independently.In 2011，the author［2］obtained a local elliptic type gradient estimate for positive solutions of equation（1.2）with 0＜p＜1. For more results of this type，see［11-18］and references therein.

Our aim is to obtain elliptic type gradient estimates for positive solutions of（1.1）and（1.2）. Precisely，we have

Theorem 1.1Let M be a Riemannian manifold of dimension n≥2 with Ricci（M）≥−k for some k≥0.Suppose that u is a positive solution to equation（1.1）in QR，T≡B（x0，R）×［t0−T，t0］⊂M×（−∞，∞）.Suppose also that u≤A in QR，T.Then，there exists a constant C=C（n）such that

Remark 1.2In［12］，using the same method but with an additional assumption q≤0，Ruan obtained

As an application，we obtain the following Liouville type theorem.

Theorem 1.3Let M be a complete，noncompact manifold of dimension n with nonnegative Ricci curvature.Suppose that q（x，t）in equation（1.1）satisfies the following conditions：

（1）q（x，t）≡q（x），that is，q is time independent；

Then，

（i）for q（x）6≡0，equation（1.1）has no positive ancient solution with logu（x，t）=o（d（x）+|t|）near infinity；

（ii）for q（x）≡0，equation（1.1）becomes heat equation，and it has only constant positive ancient solution withnear infinity.

Remark 1.4Ancient solution is a solution defined in all space and negative time.

Remark 1.5If q is a non-positive constant，then it satisfies the conditions in Theorem 1.3.

Remark 1.6The case（ii）in Theorem 1.3 has been proved by Souplet and Zhang（Theorem 1.2 in［1］）.For the aim of completeness，we still state it here.

Theorem 1.7Let M be a Riemannian manifold of dimension n≥2 with Ricci（M）≥−k for some k≥0.Suppose that u is a positive solution to equation（1.2）in QR，T≡B（x0，R）×［t0−T，t0］⊂M×（−∞，∞）.Suppose also that u≤A in QR，T.Then for any β∈（0，2），there exists a constant C=C（n，p，β）such that

As an application，we obtain the following Liouville type theorem.

Theorem 1.8Let M be a complete，noncompact manifold of dimension n with nonnegative Ricci curvature.Suppose that h（x，t）in equation（1.2）satisfies the following conditions：

（1）h（x，t）≡h（x），that is，h is time independent；

where m＞1，then

（i）for h（x）6≡0，equation（1.2）has no positive ancient solution with u（x，t）=o（［d（x）+）near infinity；

（ii）for h（x）≡0，equation（1.2）becomes heat equation，and it has only constant positive ancient solution with u（x，t）=near infinity.

Remark 1.9If h is a non-positive constant，then it satisfies the conditions in Theorem 1.8.

The remaining part of this article will be organized as follows：In Section 2，we prove Theorems 1.1 and 1.7.Theorem 1.3 and 1.8 will be proved in Section 3.

2　Proof of Theorems 1.1 and 1.7

We now derive the equation of w.

Throughout this article，we use the usual notations of covariant differentiations and the usual Einstein summation convention.A straightforward computation gives

and

By（2.2）and（2.4），we have

where Rijis the Ricci tensor of M and in the last inequality we have used the Ricci formula：fijj−fjji=Rijfjand the hypothesis Rij≥−k.

From（2.3），we know

So，we have

and

Substituting（2.6）and（2.7）into（2.5），we obtain

where，in the second inequality，we have used the fact that

For the purpose of getting the gradient estimates like what in［1］，we need to have the coefficient of w2positive.Fortunately，we have this because f≤0.

Now，we can use the cut-off function explored by Souplet and Zhang in［1］to derive the desired bounds.

Let ψ=ψ（x，t）be a smooth cut-off function supported in QR，T，satisfying the following properties：

（1）ψ=ψ（d（x，x0），t）≡ψ（r，t），ψ（r，t）=1 in0≤ψ≤1；

（2）ψ is decreasing as a radial function in the spatial variables；

Then，in view of（2.8），one has

Suppose that the maximum of ψw attained at（x1，t1）.Using the support function argument of Calabi（see［19］），we can assume，without loss of generality，that x1is not on the cut-locus of M.Then at（x1，t1），one has∆（ψw）≤0，（ψw）t≥0，and∇（ψw）=0.Therefore，

Because f≤0，we obtain

Now，we estimate all the terms on the right-hand side of（2.10）by the Young inequality. The following estimates are done at（x1，t1）.

For the first term，we have

For the second one，we have

For the third one，by the properties of ψ and the assumption on the Ricci curvature，we have

For the fourth one，we have

For the fifth one，we have

For the sixth one，we have

For the last one，we have

where q+（x，t）=max｛q（x，t），0｝.

Substituting（2.11）-（2.17）into the right-hand side of（2.10），we deduce that

Because f≤0，we obtain

It follows that for all（x，t）∈QR，T，there holds

Substituting（2.19）into（2.18），rearranging，we obtain

This ends the proof of Theorem 1.1.

Proof of Theorem 1.7We introduce the auxiliary function W=，where β is a constant to be determined.

We will derive an equation for W first.Note that

By（2.21）and（2.22），

where ǫ∈［0，1］.Here，we have used the fact ujin the last equality.

Therefore，

To get the gradient estimates as in［2，17］，we need to ensure that the coefficient of W2is positive.Fortunately，this is true for suitable β.In fact，

It is found that 2ǫ2−2ǫ+1≥So?，we can chooseto make sure the term to be positive.From now on，let

Then，we can use the cut-off function explored by Souplet and Zhang in［1］to derive the desired bounds.

Let ψ=ψ（x，t）be a smooth cut-off function supported in QR，T，satisfying the following properties：

（1）ψ=ψ（d（x，x0），t）≡ψ（r，t）；ψ（r，t）=1 in QR/2，T/2，0≤ψ≤1.

（2）ψ is decreasing as a radial function in the spatial variables.

Then，we have by（2.23），

Suppose that the maximum of ψW is reached at（x1，t1）.Using the support function argument of Calabi（see［19］），we can assume，without loss of generality，that x1is not on the cut-locus of M.Then，at（x1，t1），one has∆（ψW）≤0，（ψW）t≥0 and∇（ψW）=0.Therefore，

Now，we estimate all terms on the right-hand side of（2.25）.The main tool we will use is still the Young inequality.The following estimates are all done at（x1，t1）.

For the first term，we have

4

For the second one，we have

For the third one，we have

For the fourth one，by the properties of ψ and the assumption on the Ricci curvature，we have

For the fifth one，we have

For the sixth one，we have

where h+=max｛h，0｝，and we use the fact 2p−β＞0 because p＞1 and β＜12∈2−2∈+1≤2.

For the last one，we have

where in the third inequality we use the fact+p＞0 because of p＞1 and

Substituting（2.26）-（2.32）into the right-hand side of（2.25），we deduce that

Therefore，

Therefore，we have for all（x，t）∈QR，T，

This ends the proof of Theorem 1.7.

3　Proof of Theorems 1.3 and 1.8

Proof of Theorem 1.3Let us prove（i）first：Suppose that u（x，t）is a positive ancient solution with logu（x，t）=o（d（x）+|t|）near infinity.Fixing（x0，t0）in space-time and using Theorem 1.1 for u on the cube B（x0，R）×［t0−R2，t0］，we obtain

near infinity.Letting R→∞，it follows that|∇u（x0，t0）|=0.Because（x0，t0）is arbitrary，we get∇u≡0，so u（x，t）=u（t）.Turning to equation（1.1），we then get q（x）=constant=λ. Now，equation（1.1）changes into u′（t）=λu（t）.

From condition（2）in Theorem 1.3，we know λ≤0.We will prove that λ must be 0 and get a contradiction with the hypothesis of（i）（that is，q（x）6≡0）.In fact，suppose not，that is，λ＜0，integral u′（t）=λu（t）on（t，0］with t＜0，we get

Then，

This is a contradiction with u=near infinity.Therefore，λ=0，a contradiction.

For the proof of（ii），the reader can find it in Souplet and Zhang's article（［1］，page 1052）.

Proof of Theorem 1.8Suppose not，then let u（x，t）be an ancient solution with u（x，t）= o（［d（x）+|t|］m）near infinity.Fixing（x0，t0）in space-time，using Theorem 1.7 for u on the cube B（x0，R）×［t0−R2，t0］，for m＞1，choosing β=2），we obtain

near infinity.Letting R→∞，it follows that|∇u（x0，t0）|=0.Because（x0，t0）is arbitrary，we get∇u≡0，so u（x，t）=u（t）.Turning to equation（1.2），we then get h（x）=constant=µ. Now，equation（1.2）changes into u′（t）=µup（t）.

We prove（i）first.From condition（2）in Theorem 1.8，we knowµ≤0.We will prove that µmust be 0 and get a contradiction with the hypothesis of（i）（that is，h（x）6≡0）.In fact，suppose not，that is，µ＜0.Integating u′（t）=µup（t）on（t，0］with t＜0，we get

Then

Letting t→−∞，we have u1−p（t）＜0，which is impossible because u is a positive solution. Therefore，µ=0.Then，we get the contradiction we want.

Next，we prove（ii）.For h（x）≡0，equation（1.2）changes into the heat equation∂tu=∆u. Let u（x，t）be an ancient solution with|u（x，t）|=）near infinity.Then from Theorem 1.7，we know∇u≡0，so u（x，t）=u（t）.Taking this into∂tu=∆u，we get u′（t）=0，then u is a constant.This completes the proof of Theorem 1.8.

AcknowledgementsThe author would like to thank Professor Jiayu Li and Professor Yunyan Yang for their encouragements.

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October 13，2014；revised June 24，2015.This work is partially supported by the National Science Foundation of China（41275063 and 11401575）.