#### 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 [1–8]. 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.