The first problem of initial value on the parabolic equations of Monge-Ampère

What I'm concerned equation is
$$
-u_t det D ^ 2u = 1 quad mbox {in} Q,
$$

with the first initial value of Bounadry
$$
u (x, t) = varphi (x, t) quad mbox {on} Q partial,
$$

or $ Q $ is a non-cylindrical domain and $ Q partial is the parabolic limit.

The following conditions are given:

(1) $ Q_ {t} $ is convex, but not strictly, where $ Q_ {t_0} = Q cap {t = t_0 } $.

(2) $ varphi (x, t) in C ( overline Q) $ is convex in $ x $ and monotonous in $ t $.

Question: Is there a single, generalized solution?