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

Authors: Qiang Ru
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study the asymptotic behavior of the parabolic Monge-Ampère equation∂φ(x,t)/∂t=log(det(g(x)+Hessφ(x,t))/detg(x))−λφ(x,t) in 𝕄×(0,∞), φ(x,0)=φ0(x) in 𝕄, where 𝕄 is a compact complete Riemannian manifold, λ is a positive real parameter, and φ0(x):𝕄→ℝ is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.


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## Body

## 1. Introduction

The main purpose of this paper is to study the asymptotic behavior of the parabolic Monge-Ampère equation:(1)∂φ(x,t)∂t=log{det(g(x)+Hessφ(x,t))detg(x)}-λφ(x,t)in𝕄×(0,∞),φ(x,0)=φ0(x),in𝕄,
where 𝕄 is a compact complete Riemannian manifold, λ is a positive real parameter, and φ0(x):𝕄→ℝ 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.
Letφ be the solution of (1) with λ>0. For p>1, there exists δ>0 and f>0 depending on φ0 and ∥∇βφ∥L∞(β=0,1,2,3) such that
(2)∫𝕄‍(φp-φp¯)2dμ≤fexp(-2[(2p-1)η1p+pλ-ε(t)]t),
where φp¯ denotes the mean value of φp, η1 is the first eigenvalue of the Laplacian, and ε(t)=Δexp(-δt).Remark 2.
Ifp=1, Theorem 1 is in accordance with Theorem 1.2 of [1].Lemma 3 (see [1]).
There exists positive constantsC0(φ0,λ) and  C1 depending on (𝕄,g), φ0,  ∥φ˙∥L∞,  ∥∇β-γφ∥L∞(β=0,1,2,3;γ=1,2,3;γ≤β) such that
(3)|φ˙|≤C0exp(-λt),|∇βφ|2≤C1exp(-2λt).
Theorem 1 is proved in Section 2.


## 2. Asymptotic Behavior

Proof of Theorem1.
In local coordinates, we have the following evolution equation:(4)∂φ∂t=log{det(g+Hessφ)detg}-λφ=Δφ+logdet(gij+∇ijφ)-logdetgij-Δφ-λφ=Δφ+∫01‍dds{logdet(gij+s∇ijφ)-(s-1)Δφ(gij+s∇ijφ)}ds-λφ.
Now, setting
(5)g-sij=Δ(gij+s∇ij)-1,A=Δ∬01s|∇2φ|g-αs2dαds.
We rewrite (4) in more convenient notation as
(6)∂φ∂t=Δφ+∫01‍(g-sij-gij)∇ijφds-λφ=Δφ+∬01ddα(gij+αs∇ijφ)-1∇ijφdαds-λφ=Δφ-∬01(gik+αs∇ikφ)-1(gjl+αs∇jlφ)-1×s∇klφ∇ijφdαds-λφ=Δφ-∬01s|∇2φ|g-αs2dαds-λφ=Δφ-A-λφ.
We want to apply Gronwall inequality and hence consider the following equation:
(7)∂∂t∫𝕄‍(φp-φp¯)2dμ=2p∫𝕄‍(φp-φp¯)φp-1φ˙dμ=2p∫𝕄‍(φp-φp¯)φp-1(Δφ-A-λφ)dμ=2p∫𝕄‍(φp-φp¯)φp-1Δφdμ-2p∫𝕄‍(φp-φp¯)φp-1Adμ-2pλ∫𝕄‍(φp-φp¯)φpdμ.
Notice that
(8)∫𝕄‍φp¯(φp-φp¯)dμ=0.
We obtain
(9)-2pλ∫𝕄‍(φp-φp¯)φpdμ=-2pλ∫𝕄‍(φp-φp¯)2dμ.
Furthermore we have
(10)2p∫𝕄‍(φp-φp¯)φp-1Δφdμ=-2p∫𝕄‍∇{(φp-φp¯)φp-1}∇φdμ=-2p∫𝕄‍{∇(φp-φp¯)φp-1+(p-1)φp-2∇φ(φp-φp¯)}∇φdμ=-2∫𝕄‍|∇(φp-φp¯)|2dμ-2(p-1)p×∫𝕄‍|∇(φp-φp¯)|2dμ+2p(p-1)φp¯∫𝕄‍φp-2|∇φ|2dμ=-2(2p-1)p∫𝕄‍|∇(φp-φp¯)|2dμ+2p(p-1)φp¯∫𝕄‍φp-2|∇φ|2dμ.
We use the Poincare inequality
(11)∥∇(φp-φp¯)∥L2≥η1∥(φp-φp¯)∥L2.
It follows that
(12)2p∫𝕄‍(φp-φp¯)φp-1Δφdμ≤-2(2p-1)η1p∫𝕄‍(φp-φp¯)2dμ+2p(p-1)φp¯∫𝕄‍φp-2|∇φ|2dμ.
Moreover, we have that
(13)-2p∫𝕄‍(φp-φp¯)φp-1Adμ=2p∫01‍∫𝕄‍(g-sij-gij)∇ijφ(φp-φp¯)φp-1dμ=-2p∫01‍∫𝕄‍∇i(g-sij-gij)∇jφ(φp-φp¯)φp-1dμ-2p∫01‍∫𝕄‍(g-sij-gij)∇jφ∇i(φp-φp¯)φp-1dμ-2p∫01‍∫𝕄‍(g-sij-gij)∇jφ(φp-φp¯)∇i(φp-1)dμ=-2∫01‍∫𝕄‍∇i(g-sij-gij)∇j(φp-φp¯)(φp-φp¯)dμ-2∫01‍∫𝕄‍(g-sij-gij)∇j(φp-φp¯)∇i(φp-φp¯)dμ-2p(p-1)∫01‍∫𝕄‍(g-sij-gij)φ2(p-1)∇iφ∇jφdμ+2p(p-1)∫01‍∫𝕄‍(g-sij-gij)φp-2∇iφ∇jφφp¯dμ=-2∫01‍∫𝕄‍∇i(g-sij-gij)∇j(φp-φp¯)(φp-φp¯)dμ-2(2p-1)p∫01‍∫𝕄‍(g-sij-gij)∇j(φp-φp¯)∇i×(φp-φp¯)dμ+2p(p-1)φp¯∫01‍∫𝕄‍(g-sij-gij)φp-2∇iφ∇jφdμ≤2∫01‍∫𝕄‍|∇(g-sij-gij)|·|∇(φp-φp¯)|·|φp-φp¯|dμ+2(2p-1)p∫01‍∫𝕄‍|g-sij-gij|·|∇(φp-φp¯)|2dμ+2p(p-1)|φp¯|∫01‍∫𝕄‍|(g-sij-gij)|·|φp-2|·|∇φ|2dμ≤2Csup|∇3φ|×(∫𝕄‍|∇(φp-φp¯)|2dμ+∫𝕄‍(φp-φp¯)2dμ)+2C(2p-1)psup|∇2φ|∫𝕄‍|∇(φp-φp¯)|2dμ+2p(p-1)C|φp¯|sup|∇2φ|∫𝕄‍|φp-2|·|∇φ|2dμ.
where C is always a constant that may change from line to line.
Substituting (9), (12), and (13) in the right-hand side of (7)
(14)∂∂t∫𝕄‍(φp-φp¯)2dμ≤-2[(2p-1)η1p+pλ]∫𝕄‍(φp-φp¯)2dμ+2p(p-1)φp¯∫𝕄‍φp-2|∇φ|2dμ+2Csup|∇3φ|(∫𝕄‍|∇(φp-φp¯)|2dμ+∫𝕄‍(φp-φp¯)2dμ)+2C(2p-1)psup|∇2φ|∫𝕄‍|∇(φp-φp¯)|2dμ+2p(p-1)C|φp¯|sup|∇2φ|∫𝕄‍|φp-2|·|∇φ|2dμ.
By Lemma 3, that is, the exponential decay of |∇βφ|L∞(β=0,1,2,3), it is easy to obtain the following.
For anyε>0, there exists a T(ε) such that
(15)∂∂t∫𝕄‍(φp-φp¯)2dμ≤-2[(2p-1)η1p+pλ-ε(t)]×∫𝕄‍(φp-φp¯)2dμ+2ε(∫𝕄‍|∇(φp-φp¯)|2dμ+|φp¯|∫𝕄‍|φp-2|·|∇φ|2dμ).
The Gronwall inequality yields
(16)∫𝕄‍(φp-φp¯)2dμ≤fexp(-2[(2p-1)η1p+pλ-ε(t)]t),
where the constant f>0 depending on φ0 and ∥∇βφ∥L∞(β=0,1,2,3).
Thus, the proof of Theorem1 is completed.



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*Source: 304864-2013-03-21.xml*