Hölder Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - $\text{div}$\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time cylinder. Hölder estimates are established for the gradient of its weak solutions in the super-critical fast diffusion regime $0<p-1<q<\frac{N(p-1)}{(N-p{)}_{+}}$ where $N$ is the space dimension. Moreover, decay estimates are obtained for weak solutions and their gradient in the vicinity of possible extinction time. Two main components towards these regularity estimates are a time-insensitive Harnack inequality that is particular about this regime, and Schauder estimates for the parabolic $p$-Laplace equation.