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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 304864, 4 pages
http://dx.doi.org/10.1155/2013/304864
Research Article

A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds

Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

Received 8 November 2012; Accepted 21 February 2013

Academic Editor: Junjie Wei

Copyright © 2013 Qiang Ru. 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 study the asymptotic behavior of the parabolic Monge-Ampère equation in , in , where is a compact complete Riemannian manifold, λ is a positive real parameter, and is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.

1. Introduction

The main purpose of this paper is to study the asymptotic behavior of the parabolic Monge-Ampère equation: where is a compact complete Riemannian manifold, is a positive real parameter, and is a smooth function. We show a meaningful precisely asymptotic result which is more general than those in [1].

Monge-Ampère equations arise naturally from some problems in differential geometry. The existence and regularity of solutions to Monge-Ampère equations have been investigated by many mathematicians [18]. The long time existence and convergence of solution to (1) have been investigated in [1]. To some extent, we extend asymptotic result obtained in [1] in this paper. Hence, our main result is following analogue of Theorem 1.2 of [1].

Theorem 1. Let be the solution of (1) with . For , there exists and depending on and such that where denotes the mean value of , is the first eigenvalue of the Laplacian, and .

Remark 2. If , Theorem 1 is in accordance with Theorem 1.2 of [1].

Lemma 3 (see [1]). There exists positive constants and depending on , ,  , such that Theorem 1 is proved in Section 2.

2. Asymptotic Behavior

Proof of Theorem 1. In local coordinates, we have the following evolution equation: Now, setting We rewrite (4) in more convenient notation as We want to apply Gronwall inequality and hence consider the following equation: Notice that We obtain Furthermore we have We use the Poincare inequality It follows that Moreover, we have that where is always a constant that may change from line to line.
Substituting (9), (12), and (13) in the right-hand side of (7) By Lemma 3, that is, the exponential decay of , it is easy to obtain the following.
For any , there exists a such that The Gronwall inequality yields where the constant depending on and .
Thus, the proof of Theorem 1 is completed.

References

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