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?