Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds
© Zhao; licensee Springer 2013
Received: 31 May 2013
Accepted: 9 August 2013
Published: 27 August 2013
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 , .
KeywordsLichnerowicz equation positive solutions Harnack inequality
where h, p, q, A, B are real constants and , .
where , c are two real constants. More precisely, the author  obtained the following result.
Theorem 1.1 (Yang )
- (i)If , then satisfies the estimate
- (ii)If , then satisfies the estimate
on , where and are some universal constants independent of geometry of M.
on manifold and obtained the Harnack differential inequality.
Theorem 1.2 (Song and Zhao )
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.
- (1)if , , we have
- (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).
- (1)if , , we have
- (2)if , , we get
δ are positive constants with and .
Let , in Corollary 1.4, we get a Li-Yau-type gradient estimate.
As an application, we have the following Harnack inequality.
where , .
2 Proof of Theorem 1.3
which completes the proof of Lemma 2.1. □
In inequality (2.5), if , then △φ can be controlled by , so in any case, , where c is some positive constant.
- (1)if , , then we have
- (2)if , , then we have
Similarly, we can obtain (2) of Theorem 1.3.
where , . □
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).
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