Gradient regularity for a class of doubly nonlinear parabolic partial differential equations

In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type align* ₜ uq - div\, A (x, t, Du) =0 T, align* with q>0, T= (0, T) ^n+1 a space-time cylinder, and A=A (x, t, ) a vector field satisfying standard p-growth conditions. Our main result establishes the local H\"older continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime 0<p-1<q<n (p-1) (n-p) _+. This result is achieved by utilizing a time-insensitive Harnack inequality and Schauder estimates that are developed for equations of parabolic p-Laplacian type. Additionally, we establish a local L^-bound for the spatial gradient.