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Research Article | Open Access

Volume 2013 |Article ID 304864 | 4 pages | https://doi.org/10.1155/2013/304864

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

Accepted21 Feb 2013
Published21 Mar 2013

#### 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 .

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 . The long time existence and convergence of solution to (1) have been investigated in . To some extent, we extend asymptotic result obtained in  in this paper. Hence, our main result is following analogue of Theorem 1.2 of .

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 .

Lemma 3 (see ). 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.

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