<p>This paper aims at providing asymptotic behaviors at infinity of solutions to the parabolic Monge-Ampère equations <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1992_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$-u_t\text {det}D_x^2u=f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> <mtext>det</mtext> <msubsup> <mi>D</mi> <mi>x</mi> <mn>2</mn> </msubsup> <mi>u</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1992_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^{n+1}_-$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>-</mo> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1992_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1992_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$-u_t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> has a positive upper and lower bound and <i>f</i> tends to 1 at infinity. We obtain results that are quite different from the elliptic case, and also improve the existing works of the parabolic case. There are counterexamples to illustrate that the asymptotic behavior is optimal.</p>

Asymptotic Behaviors at Infinity of Ancient Solutions to the Parabolic Monge-Ampère Equation

期刊论文
发表日期：

2025年3月28日

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Kui Yan,
Jiguang Bao

摘要

This paper aims at providing asymptotic behaviors at infinity of solutions to the parabolic Monge-Ampère equations $-u_t\text {det}D_x^2u=f$ $- u_{t} det D_{x}^{2} u = f$ in $\mathbb {R}^{n+1}_-$ $R_{-}^{n + 1}$ for $n\ge 1$ $n \geq 1$ , where $-u_t$ $- u_{t}$ has a positive upper and lower bound and f tends to 1 at infinity. We obtain results that are quite different from the elliptic case, and also improve the existing works of the parabolic case. There are counterexamples to illustrate that the asymptotic behavior is optimal.