Hölder gradient estimates for a class of singular or degenerate parabolic equations

We prove interior Hölder estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation $$ u_t=|\nabla u|^κ\mbox{div} (|\nabla u|^{p-2}\nabla u), $$ where $p\in (1,\infty)$ and $κ\in (1-p,\infty).$ This includes the from $L^\infty$ to $C^{1,α}$ regularity for parabolic $p$-Laplacian equations in both divergence form with $κ=0$, and non-divergence form with $κ=2-p$. This work is a continuation of a paper by the last two authors \cite{JS}.

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