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 u^q - div\, A(x,t,Du)=0 ΩT, {align*} with $q>0$, ΩT=Ω×(0,T)ⁿ⁺¹ a space-time cylinder, and A=A(x,t,ξ) 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.