$C^{1,\text{al}}$ regularity of solutions to parabolic Monge-Amp\'ere equations

We study interior $C^{1,\text{al}}$ regularity of viscosity solutions of the parabolic Monge-Amp'ere equation $$u_t=b(x,t)\text{ddua},$$ with exponent $p>0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $\frac{1}{n-2}$ then solutions become instantly $C^{1,\text{al}}$ 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 $C^{1,\beta }$.