## Boundary Value Problems

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# Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds

Boundary Value Problems20132013:190

DOI: 10.1186/1687-2770-2013-190

Accepted: 9 August 2013

Published: 27 August 2013

## Abstract

In this paper, we study the gradient estimates for positive solutions to the following parabolic Lichnerowicz equations

on complete noncompact Riemannian manifolds, where h, p, q, A, B are real constants and , .

MSC:58J05, 58J35.

### Keywords

Lichnerowicz equation positive solutions Harnack inequality

## 1 Introduction

Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we study the following nonlinear parabolic equation
(1.1)

where h, p, q, A, B are real constants and , .

Gradient estimates play an important role in the study of PDE, especially the Laplace equation and heat equation. Li [1] derived the gradient estimates and Harnack inequalities for positive solutions of nonlinear equations and on Riemannian manifolds. The author in [1] also obtained a theorem of Liouville-type for positive solutions of the nonlinear elliptic equation. Later, Yang [2] gave the gradient estimates for the solution to the elliptic equation with singular nonlinearity
(1.2)

where , c are two real constants. More precisely, the author [2] obtained the following result.

Theorem 1.1 (Yang [2])

Let M be a noncompact complete Riemannian manifold of dimension n without boundary. Let be a geodesic ball of radius 2R around . We denote , with , to be a lower bound of the Ricci curvature on , i.e., for all tangent field ξ on . Suppose that is a positive smooth solution of the equation (1.2) with , c being two real constants. Then we have:
1. (i)
If , then satisfies the estimate

on , where and are some universal constants independent of geometry of M.
1. (ii)
If , then satisfies the estimate

on , where and are some universal constants independent of geometry of M.

For some interesting gradient estimates in this direction, we can refer to [37].

Recently, Song and Zhao [8] studied a generalized elliptic Lichnerowicz equation
(1.3)
on compact manifold . The authors in [8] got the local gradient estimate for the positive solutions of (1.3). Moreover, they considered the following parabolic Lichnerowicz equation
(1.4)

on manifold and obtained the Harnack differential inequality.

Theorem 1.2 (Song and Zhao [8])

Let M be a compact Riemannian manifold without boundary, . Let . Assume that is any positive solution of (1.4) on M with , , and . Denote , suppose that , , , . If with at , then we have

While the author considered the gradient estimates on compact Riemannian manifolds in Theorem 1.2, it is natural to study this problem on complete noncompact manifolds. Motivated by the work above, we present our main results as follows.

Theorem 1.3 Let be a complete noncompact n-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant , where and in the metric ball around . Assume that u is a positive solution of (1.1) with for all . Then
1. (1)
if , , we have

2. (2)
if , , we get

where , c, , , δ are positive constants with and .

Let , we can get the following global gradient estimates for the nonlinear parabolic equation (1.1).

Corollary 1.4 Let be a complete noncompact n-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant , where . Assume that u is a positive solution of (1.1) with for all . Then
1. (1)
if , , we have

2. (2)
if , , we get

δ are positive constants with and .

Let , in Corollary 1.4, we get a Li-Yau-type gradient estimate.

Corollary 1.5 Let be a complete noncompact n-dimensional Riemannian manifold with . Assume that is a positive solution to the equation
on complete noncompact manifolds, where h, q, B are real constants and . Then we have
(1.5)

As an application, we have the following Harnack inequality.

Theorem 1.6 Let be a complete noncompact n-dimensional Riemannian manifold with . Assume that is a positive solution to the equation
on complete noncompact manifolds, where h, q, B are real constants and , . Then for any points and on with , we have the following Harnack inequality:

where , .

## 2 Proof of Theorem 1.3

Assume that u is a positive solution to (1.1). Set , then w satisfies the equation
(2.1)
Lemma 2.1 Let be a complete noncompact n-dimensional Riemannian manifold with Ricci curvature bounded from below by the constant , where and in the metric ball around . Let w be a positive solution of (2.1), then
where

and .

Proof Define
where . By the Bochner formula, we have
(2.2)
By a direct computation, we have
(2.3)
and
and we know
Therefore, by equalities (2.2) and (2.3), we obtain
This implies that,
and
Therefore, it follows that

which completes the proof of Lemma 2.1. □

We take a cut-off function defined on such that for , for , and . Furthermore, satisfies
and
for some absolute constants . Denote by the distance between x and p in M. Set
Using an argument of Cheng and Yau [9], we can assume that with support in . Direct calculation shows that on
(2.4)
By the Laplacian comparison theorem in [10],
(2.5)

In inequality (2.5), if , then φ can be controlled by , so in any case, , where c is some positive constant.

For , let be a point in , at which φF attains its maximum value P, and we assume that P is positive (otherwise the proof is trivial). At the point , we have
It follows that
This inequality, together with inequalities (2.4) and (2.5), yields
where
At , by Lemma 2.1, we have
it follows that
here we used
Then we have
Next, we consider the following two cases:
1. (1)
if , , then we have

multiplying both sides of the inequality above by , we have
So, it follows that
Since
we get
Now, (1) of Theorem 1.3 can be easily deduced from the inequality above;
1. (2)
if , , then we have

multiplying both sides of the inequality above by , we have
So, it follows that

Similarly, we can obtain (2) of Theorem 1.3.

Proof of Theorem 1.6 For any points and on with , we take a curve parameterized with and . One gets from Corollary 1.5 that
which means that
Therefore,

where , . □

## Declarations

### Acknowledgements

The author would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that improved the quality of the paper. Moreover, the author would like to thank his supervisor Professor Kefeng Liu for his constant encouragement and help. This work is supported by the Postdoctoral Science Foundation of China (2013M531342) and the Fundamental Research Funds for the Central Universities (NS2012065).

## Authors’ Affiliations

(1)
Department of Mathematics, Nanjing University of Aeronautics and Astronautics

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