$C^{1,\alpha }$-regularity for solutions of degenerate/singular fully nonlinear parabolic equations

We establish the interior $C^{1,\alpha }$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t=\mathrm{\mid }Du{\mathrm{\mid }}^{\gamma }F(D^{2}u)+f\mathrm{.}$$ For this purpose, we prove the well-posedness of the regularized Dirichlet problem \begin{equation*} \left{ \begin{aligned} u_t&=(1+|Du|^2)^{γ/2}F(D^2u) &&$\text{in }{Q}_1$ \newline u&=\varphi &&$\text{on }{\mathrm{\partial }}_p{Q}_1$. \end{aligned}\right. \end{equation*} Our approach utilizes the Bernstein method with approximations in view of difference quotient.