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

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 equationon 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 equationis 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 flowHsu 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) inwith for some positive constant and . Then there exists a constant depending only on the dimension of such thatfor 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) inwith for some positive constant and . Then there exists a constant depending only on the dimension of such that  if , thenfor all with , where ;  if , thenfor all with , where ;  if , thenfor 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 thatwhere .

Proof. Under the scaling , we have . From (1), we obtain that satisfies the following equation:where constant satisfies . Let and . Then we haveUsing (3), we can obtainBy the definition of , we haveOn the other hand,where, in the second equality, we used the Ricci formulaBy the formulas (18) and (19), we arrive atNote thatTherefore, (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 thatNow 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 deduceat . Next, we will find an upper bound for each term of the right-hand side of (25). By the Cauchy inequality, we haveIt has been shown in [20] (see formulas (3.30), (3.32), and (3.34) in [20]) thatPutting (26)-(27) into (25), we obtainSince , the inequality (28) impliesTherefore, we deduce that, for any ,from . Noticing in and , we have the fact thatholds at any , which shows thatWe 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 thatwhere .

Proof. By the scaling , we have . Therefore, we obtain from (9) that satisfieswhere . Let andThen, the function satisfiesUsing (3) again, one hasIt follows from (35) thatSimilarly, by the Ricci formula, we obtainThus, we derive from (38) and (39) the following:Using the relationship (40) can be written aswhere is a function depending on which will be determined. Applying the inequality into (42) givesTaking in (44), we deriveThus, 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 thatNow 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 thatat .
Case One (). Note thatIn this case, we have and . Hencewhere . Putting (50), (26)-(27) into (48), we obtainTherefore, we deduce, for any , thatfrom . Noticing in and , we have the fact thatholds at any , which shows thatCase Two (). In this case, we have and . Hencewhere . Similarly, putting (55), (26)-(27) into (48), we obtainThus, we can obtain thatholds at any , which shows thatCase Three . In this case, we also have . HencePutting (59), (26)-(27) into (48), we obtainBecause of , we can obtain thatholds at any , which shows thatWe 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).