Ancient solutions of exterior problem of parabolic Monge–Ampère equations

We use Perron method to prove the existence of ancient solutions of exterior problem for a kind of parabolic Monge–Ampere equation $$-\,u_t\det D^2u=f$$ with prescribed asymptotic behavior at infinity outside some certain bowl-shaped domain in the lower half space for $$n\ge 3$$ , where f is a perturbation of 1 at infinity. We raise this problem for the first time and construct a new subsolution to it. We also use similar method to prove the existence of the entire solutions.