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We prove the existence and regularity of convex solutions to the first initial-boundary value problem of the parabolic Monge-Amp\`ere equation \{eqnarray &uₜ= D²u in QT, \\ &u= on ₚQT, eqnarray. where is a smooth function, QT= (0, T], ₚ QT is the parabolic boundary of QT, and is a uniformly convex domain in Rⁿ with smooth boundary. Our approach can also be used to prove similar results for -Gauss curvature flow with any 0< 1.
Zhou et al. (Mon,) studied this question.