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?