C1,α regularity of solutions to parabolic Monge-Ampére equations

We study interior $C^{1, \alpha}$ regularity of viscosity solutions of the parabolic Monge-Amp\'ere equation $$ u_t = b(x,t)\,\big(\!\det D^2 u\big)^p, $$ with exponent $p >0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $1\over{n-2}$ then solutions become instantly $C^{1,\alpha}$ in the interior. Also, we prove the same result for any power $p>0$ at those points where either the solution separates from the initial data, or where the initial data is $C^{1,\beta}$.