Entire solutions to the parabolic Monge--Ampère equation with unbounded nonlinear growth in time

The Liouville type theorem on the parabolic Monge–Ampère equation $-u_t\det D^{2}u=1$ states that any entire parabolically convex classical solution must be of form $-t+\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}2$ up to a re-scaling and transformation, under additional assumption that partial derivative with respect to time variable $u_t$ is strictly negative and bounded. In this paper, we study the case when $u_t$ is unbounded, prove an existence result of entire parabolically convex smooth solution and investigate the asymptotic behavior near infinity.