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

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