Optimal Asymptotic Behavior at Infinity of Ancient Solution to the Parabolic Monge-Amp\`ere Equation with Slow Perturbation Term

TL;DR

This paper analyzes the long-term asymptotic behavior of solutions to a parabolic Monge-Ampère equation with a slowly converging right-hand side, extending elliptic estimates to the parabolic case.

Contribution

It establishes the optimal asymptotic behavior of ancient solutions to the parabolic Monge-Ampère equation with slow perturbation, extending previous elliptic results.

Findings

Derived the optimal asymptotic behavior at infinity.

Extended elliptic estimates to the parabolic setting.

Provided conditions for solutions with slow convergence.

Abstract

In this paper, we obtain optimal asymptotic behavior of parabolically convex $C^{2,1}$ solution to the parabolic Monge-Amp\`ere 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.