Optimal regularity for degenerate parabolic equations on a flat boundary

We establish the optimal regularity of viscosity solutions to \begin{equation*} u_t - x_n^γΔu = f, \end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero Dirichlet boundary condition on $\{x_n=0\}$, the optimal regularity of $u$ up to the flat boundary $\{x_n=0\}$ is $C^{1,1-\gamma }$. Moreover, for the homogeneous equations, we establish that the optimal regularity of $u$ is $C^{2,1-\gamma }$ in the spatial variables, and that $x_n^{-\gamma }u$ is smooth in the variables $x^{\mathrm{\prime }}$ and $t$.