JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 304864 10.1155/2013/304864 304864 Research Article A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds Ru Qiang Wei Junjie Center of Mathematical Sciences Zhejiang University Hangzhou 310027 China zju.edu.cn 2013 21 03 2013 2013 08 11 2012 21 02 2013 2013 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.

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 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: (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 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 φ be the solution of (1) with λ>0. For p>1, there exists δ>0 and f>0 depending on φ0 and βφL(β=0,1,2,3) such that (2)𝕄(φp-φp¯)2dμfexp(-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.

If p=1, Theorem 1 is in accordance with Theorem 1.2 of .

Lemma 3 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

There exists positive constants C0(φ0,λ) and C1 depending on (𝕄,g), φ0,  φ˙L, β-γφL(β=0,1,2,3;γ=1,2,3;γβ) such that (3)|φ˙|C0exp(-λt),|βφ|2C1exp(-2λt). Theorem 1 is proved in Section 2.

2. Asymptotic Behavior Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

In local coordinates, we have the following evolution equation: (4)φt=log{det(g+Hessφ)detg}-λφ=Δφ+logdet(gij+ijφ)-logdetgij-Δφ-λφ=Δφ+01dds{logdet(gij+sijφ)-(s-1)Δφ(gij+sijφ)}ds-λφ. Now, setting (5)g-sij=Δ(gij+sij)-1,A=Δ01s|2φ|g-αs2dαds. We rewrite (4) in more convenient notation as (6)φt=Δφ+01(g-sij-gij)ijφds-λφ=Δφ+01ddα(gij+αsijφ)-1ijφdαds-λφ=Δφ-01(gik+αsikφ)-1(gjl+αsjlφ)-1×sklφijφdαds-λφ=Δφ-01s|2φ|g-αs2dαds-λφ=Δφ-A-λφ. We want to apply Gronwall inequality and hence consider the following equation: (7)t𝕄(φp-φp¯)2dμ=2p𝕄(φp-φp¯)φp-1φ˙dμ=2p𝕄(φp-φp¯)φp-1(Δφ-A-λφ)dμ=2p𝕄(φp-φp¯)φp-1Δφdμ-2p𝕄(φp-φp¯)φp-1Adμ-2pλ𝕄(φp-φp¯)φpdμ. Notice that (8)𝕄φp¯(φp-φp¯)dμ=0. We obtain (9)-2pλ𝕄(φp-φp¯)φpdμ=-2pλ𝕄(φp-φp¯)2dμ. Furthermore we have (10)2p𝕄(φp-φp¯)φp-1Δφdμ=-2p𝕄{(φp-φp¯)φp-1}φdμ=-2p𝕄{(φp-φp¯)φp-1+(p-1)φp-2φ(φp-φp¯)}φdμ=-2𝕄|(φp-φp¯)|2dμ-2(p-1)p×𝕄|(φp-φp¯)|2dμ+2p(p-1)φp¯𝕄φp-2|φ|2dμ=-2(2p-1)p𝕄|(φp-φp¯)|2dμ+2p(p-1)φp¯𝕄φp-2|φ|2dμ. We use the Poincare inequality (11)(φp-φp¯)L2η1(φp-φp¯)L2. It follows that (12)2p𝕄(φp-φp¯)φp-1Δφdμ-2(2p-1)η1p𝕄(φp-φp¯)2dμ+2p(p-1)φp¯𝕄φp-2|φ|2dμ. Moreover, we have that (13)-2p𝕄(φp-φp¯)φp-1Adμ=2p01𝕄(g-sij-gij)ijφ(φp-φp¯)φp-1dμ=-2p01𝕄i(g-sij-gij)jφ(φp-φp¯)φp-1dμ-2p01𝕄(g-sij-gij)jφi(φp-φp¯)φp-1dμ-2p01𝕄(g-sij-gij)jφ(φp-φp¯)i(φp-1)dμ=-201𝕄i(g-sij-gij)j(φp-φp¯)(φp-φp¯)dμ-201𝕄(g-sij-gij)j(φp-φp¯)i(φp-φp¯)dμ-2p(p-1)01𝕄(g-sij-gij)φ2(p-1)iφjφdμ+2p(p-1)01𝕄(g-sij-gij)φp-2iφjφφp¯dμ=-201𝕄i(g-sij-gij)j(φp-φp¯)(φp-φp¯)dμ-2(2p-1)p01𝕄(g-sij-gij)j(φp-φp¯)i×(φp-φp¯)dμ+2p(p-1)φp¯01𝕄(g-sij-gij)φp-2iφjφdμ201𝕄|(g-sij-gij)|·|(φp-φp¯)|·|φp-φp¯|dμ+2(2p-1)p01𝕄|g-sij-gij|·|(φp-φp¯)|2dμ+2p(p-1)|φp¯|01𝕄|(g-sij-gij)|·|φp-2|·|φ|2dμ2Csup|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)t𝕄(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 any ε>0, there exists a T(ε) such that (15)t𝕄(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¯)2dμfexp(-2[(2p-1)η1p+pλ-ε(t)]t), where the constant f>0 depending on φ0 and βφL  (β=0,1,2,3).

Thus, the proof of Theorem 1 is completed.

Huisken B. Parabolic Monge-Ampère equations on Riemannian manifolds Journal of Functional Analysis 1997 147 1 140 163 10.1006/jfan.1996.3062 MR1453179 ZBL0895.58053 Aubin T. Nonlinear Analysis on Manifolds, Monge-Ampère Equations 1982 252 New York, NY, USA Springerg Fundamental Principles of Mathematical Sciences 10.1007/978-1-4612-5734-9 MR681859 Delanoë P. Équations du type Monge-Ampère sur les variétés riemanniennes compactes I Journal of Functional Analysis 1981 40 3 358 386 10.1016/0022-1236(81)90054-9 MR611589 ZBL0466.58029 Delanoë P. Équations du type de Monge-Ampère sur les variétés riemanniennes compactes II Journal of Functional Analysis 1981 41 3 341 353 10.1016/0022-1236(81)90080-X MR619957 ZBL0474.58023 Delanoë P. Équations du type de Monge-Ampère sur les variétés riemanniennes compactes III Journal of Functional Analysis 1982 45 3 403 430 10.1016/0022-1236(82)90013-1 MR650189 ZBL0497.58026 Songzhe L. Existence of solutions to initial value problem for a parabolic Monge-Ampère equation and application Nonlinear Analysis: Theory, Methods & Applications 2006 65 1 59 78 10.1016/j.na.2005.05.047 MR2226259 ZBL1100.35047 Zhang Z. T. Wang K. L. Existence and non-existence of solutions for a class of Monge-Ampère equations Journal of Differential Equations 2009 246 7 2849 2875 10.1016/j.jde.2009.01.004 MR2503025 ZBL1165.35023 Huang J. N. Duan Z. W. Existence of the global solution for the parabolic Monge-Ampère equations on compact Riemannian manifolds Journal of Mathematical Analysis and Applications 2012 389 1 597 607 10.1016/j.jmaa.2011.12.005 MR2876524 ZBL1237.58027