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Optimal Asymptotic Behavior at Infinity of Ancient Solution to the Parabolic Monge-Ampère Equation with Slow Perturbation Term

Kui Yan, Jiguang Bao

Abstract

In this paper, we obtain optimal asymptotic behavior of parabolically convex $C^{2,1}$ solution to the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f$ converges to $1$ at infinity with a slow rate. This result extends the elliptic estimate in \cite{lb5} to the parabolic setting.

Optimal Asymptotic Behavior at Infinity of Ancient Solution to the Parabolic Monge-Ampère Equation with Slow Perturbation Term

Abstract

In this paper, we obtain optimal asymptotic behavior of parabolically convex solution to the parabolic Monge-Ampère equation , where converges to at infinity with a slow rate. This result extends the elliptic estimate in \cite{lb5} to the parabolic setting.

Paper Structure

This paper contains 4 sections, 3 theorems, 112 equations.

Table of Contents

  1. introduction and main results
  2. a slow convergence solution to $\left(D_t-\Delta_x\right)u=f$
  3. smoothing of solution
  4. proof of Theorem \ref{['mainthm1']}

Key Result

Theorem 1.1

For $n\ge1$, let $u\in C^{2,1}(\mathbb{R}^{n+1}_{-})$ be a parabolically convex solution to pma1 with ut, where $f\in C^0(\mathbb{R}^{n+1}_{-})$ satisfies f for $m\ge3$ and $\beta\in(0,2]$. Then $u\in C^{2m,m}$ outside a bounded set in $\mathbb{R}^{n+1}_{-}$. Additionally there exist a $n\times n$ s where

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • proof : Case for $0<\beta\le 1$
  • proof : Case for $1<\beta\le 2$