Abstract
Keywords:
Lichnerowicz equation; positive solutions; Harnack inequality1 Introduction
Let M be an ndimensional complete noncompact Riemannian manifold. In this paper, we study the following nonlinear parabolic equation
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 Liouvilletype 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
where , c are two real constants. More precisely, the author [2] obtained the following result.
Theorem 1.1 (Yang [2])
LetMbe a noncompact complete Riemannian manifold of dimensionnwithout boundary. Letbe a geodesic ball of radius 2Raround. We denote, with, to be a lower bound of the Ricci curvature on, i.e., for all tangent fieldξon. Suppose thatis a positive smooth solution of the equation (1.2) with, cbeing two real constants. Then we have:
(i) If, thensatisfies the estimate
on, whereandare some universal constants independent of geometry ofM.
(ii) If, thensatisfies the estimate
on, whereandare some universal constants independent of geometry ofM.
For some interesting gradient estimates in this direction, we can refer to [37].
Recently, Song and Zhao [8] studied a generalized elliptic Lichnerowicz equation
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
on manifold and obtained the Harnack differential inequality.
Theorem 1.2 (Song and Zhao [8])
LetMbe a compact Riemannian manifold without boundary, . Let. Assume thatis any positive solution of (1.4) onMwith, , and. Denote, suppose that, , , . Ifwithat, 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 1.3Letbe a complete noncompactndimensional Riemannian manifold with Ricci tensor bounded from below by the constant, whereandin the metric ballaround. Assume thatuis a positive solution of (1.1) withfor all. Then
where, c, , , δare positive constants withand.
Let , we can get the following global gradient estimates for the nonlinear parabolic equation (1.1).
Corollary 1.4Letbe a complete noncompactndimensional Riemannian manifold with Ricci tensor bounded from below by the constant, where. Assume thatuis a positive solution of (1.1) withfor all. Then
δare positive constants withand.
Let , in Corollary 1.4, we get a LiYautype gradient estimate.
Corollary 1.5Letbe a complete noncompactndimensional Riemannian manifold with. Assume thatis a positive solution to the equation
on complete noncompact manifolds, whereh, q, Bare real constants and. Then we have
As an application, we have the following Harnack inequality.
Theorem 1.6Letbe a complete noncompactndimensional Riemannian manifold with. Assume thatis a positive solution to the equation
on complete noncompact manifolds, whereh, q, Bare real constants and, . Then for any pointsandonwith, we have the following Harnack inequality:
2 Proof of Theorem 1.3
Assume that u is a positive solution to (1.1). Set , then w satisfies the equation
Lemma 2.1Letbe a complete noncompactndimensional Riemannian manifold with Ricci curvature bounded from below by the constant, whereandin the metric ballaround. Letwbe a positive solution of (2.1), then
where
Proof Define
where . By the Bochner formula, we have
By a direct computation, we have
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 cutoff 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
By the Laplacian comparison theorem in [10],
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
it follows that
here we used
Following Davies [11] (see also Negrin [12]), we set
Then we have
Next, we consider the following two cases:
multiplying both sides of the inequality above by sφ, we have
So, it follows that
Since
we get
Now, (1) of Theorem 1.3 can be easily deduced from the inequality above;
multiplying both sides of the inequality above by sφ, 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,
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author completed the paper. The author read and approved the final manuscript.
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).
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