Journal of Function Spaces
Volume 2016 (2016), Article ID 2894207, 7 pages
http://dx.doi.org/10.1155/2016/2894207

Research Article

Hamilton-Souplet-Zhang’s Gradient Estimates for Two Types of Nonlinear Parabolic Equations under the Ricci Flow

Guangyue Huang¹ and Bingqing Ma²

¹College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China
²Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China

Received 10 November 2015; Accepted 29 December 2015

Academic Editor: Igor E. Verbitsky

Copyright © 2016 Guangyue Huang and Bingqing Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider gradient estimates for two types of nonlinear parabolic equations under the Ricci flow: one is the equation with being two real constants; the other is with being two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang-type gradient estimates.

1. Introduction

Since the nonlinear parabolic equation

on a given Riemannian manifold is related to gradient Ricci solitons which are self-similar solutions to the Ricci flow, many attentions are paid to the study on gradient estimates for (1); for example, see [1–7]. Here in (1) are two real constants. Clearly, a heat equation

is a special case of (1) when . Hence many known results on heat equations are generalized to the nonlinear parabolic equation (1). For gradient estimates of solutions to (1) of Li-Yau type, Davies type, Hamilton type, and Li-Xu type on a given Riemannian manifold, we refer to [1, 3, 7, 8] and the references therein. In a recent paper [9], Dung and Khanh obtained Hamilton-Souplet-Zhang-type gradient estimates on a given Riemannian manifold for (1). On a family of Riemannian metrics evolving from the Ricci flow

Hsu in [10] obtained Li-Yau-type gradient estimates of (1).

In [11], generalizing Hamilton’s estimate in [12], Souplet and Zhang proved the following result.

Theorem A (see [11]). Let be an -dimensional Riemannian manifold with , where is a nonnegative constant. Suppose that is a positive solution to (2) in with . Then in ,

where constant depends only on the dimension .

The key to prove Theorem A of Souplet and Zhang is the scaling . After this scaling, (2) becomes the following heat equation with respect to :

since the heat equation is linear. Under this case, we obtain that . Inspired by this method, in this paper, we also adopt the similar scaling method by to study the nonlinear parabolic equation (1). By the scaling, we can derive from (1) the following analogous equation:

where constant satisfies . That is, we only need to study the nonlinear equation (6) with .

Our first result is the following Hamilton-Souplet-Zhang-type gradient estimates of the nonlinear equation (1) under the Ricci flow.

Theorem 1. Let be a complete Riemannian manifold with a family of Riemannian metrics evolving from the Ricci flow (3). Suppose that is a positive solution to (1) in

with for some positive constant and . Then there exists a constant depending only on the dimension of such that

for all with , where .

The study of Li-Yau-type estimates of the following nonlinear parabolic equation:

where are two real constants, can be traced back to Li [13]. Later, for , Zhu in [14] obtained Hamilton-Souplet-Zhang-type gradient estimates of (9) on a given Riemannian manifold. On gradient estimates of the elliptic case of (9), see [15, 16]. A natural subject is to study Hamilton-Souplet-Zhang-type gradient estimates of the nonlinear equation (9) under the Ricci flow. Our second result is the following.

Theorem 2. Let be a complete Riemannian manifold with a family of Riemannian metrics evolving from the Ricci flow (3). Suppose that is a positive solution to (9) in

with for some positive constant and . Then there exists a constant depending only on the dimension of such that

  if , then

for all with , where ;  if , then

for all with , where ;  if , then

for all with .

Remark 3. Taking in (8), we obtain the estimate (16) of Theorem  2.2 in [17] with respect to the heat equation under the Ricci flow. Hence, our estimates in Theorem 1 extend Bailesteanu, Cao, and Pulemotov’s Theorem 2.2.

Remark 4. There are many studies on gradient estimates of the heat equation (2) under geometric flows; we refer to [18, 19] among others.

2. Proof of Theorem 1

In order to prove our Theorem 1, we first give a lemma which will play an important role in the proof.

Lemma 5. Let be a complete Riemannian manifold with a family of Riemannian metrics evolving from the Ricci flow (3). Let be a positive solution to (1) with . Denoting by , and Then, it holds that

where .

Proof. Under the scaling , we have . From (1), we obtain that satisfies the following equation:

where constant satisfies . Let and . Then we have

Using (3), we can obtain

By the definition of , we have

On the other hand,

where, in the second equality, we used the Ricci formula

By the formulas (18) and (19), we arrive at

Note that

Therefore, (21) can be written as

Then, the desired estimate (14) follows.

Proof of Theorem 1. Let be a smooth cutoff function supported in which satisfies the following properties (see page  13 in [20] or Lemma  2.1 in [17]):

(1); in and .(2) is decreasing as a radial function in the spatial variables.(3) and for every .(4).

We use (14) to conclude that

Now we let be a maximum point of in the closure of , and where (otherwise the proof is trivial). Then at the point , we have , , and . Thus, from (24), we deduce

at . Next, we will find an upper bound for each term of the right-hand side of (25). By the Cauchy inequality, we have

It has been shown in [20] (see formulas (3.30), (3.32), and (3.34) in [20]) that

Putting (26)-(27) into (25), we obtain

Since , the inequality (28) implies

Therefore, we deduce that, for any ,

from . Noticing in and , we have the fact that

holds at any , which shows that

We complete the proof of the estimate (8) in Theorem 1.

3. Proof of Theorem 2

As in the proof of Theorem 1, we first give a key lemma.

Lemma 6. Let be a complete Riemannian manifold with a family of Riemannian metrics evolving from the Ricci flow (3). Let be a positive solution to (9) with . Denoting by , and . Then, it holds that

where .

Proof. By the scaling , we have . Therefore, we obtain from (9) that satisfies

where . Let and

Then, the function satisfies

Using (3) again, one has

It follows from (35) that

Similarly, by the Ricci formula, we obtain

Thus, we derive from (38) and (39) the following:

Using the relationship

(40) can be written as

where is a function depending on which will be determined. Applying the inequality

into (42) gives

Taking

in (44), we derive

Thus, the desired estimate (33) is attained.

Proof of Theorem 2. Let be a smooth cutoff function supported in which is defined in Section 2. We use (33) to conclude that

Now we let be a maximum point of in the closure of , where (otherwise the proof is trivial). Then at the point , we have , , and . Thus, from (47), we deduce that

at .
Case One (). Note that

In this case, we have and . Hence

where . Putting (50), (26)-(27) into (48), we obtain

Therefore, we deduce, for any , that

from . Noticing in and , we have the fact that

holds at any , which shows that

Case Two (). In this case, we have and . Hence

where . Similarly, putting (55), (26)-(27) into (48), we obtain

Thus, we can obtain that

holds at any , which shows that

Case Three . In this case, we also have . Hence

Putting (59), (26)-(27) into (48), we obtain

Because of , we can obtain that

holds at any , which shows that

We complete the proof of Theorem 2.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank an anonymous referee for his or her careful reading and helpful comments which made this paper more readable. The research of the first author is supported by NSFC (nos. 11371018 and 11171091), Henan Province Backbone Teacher (no. 2013GGJS-057), and IRTSTHN (14IRTSTHN023). The research of the second author is supported by NSFC (no. 11401179) and Henan Provincial Education Department (no. 14B110017).

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