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

Abstract

In this paper, we study the local gradient regularity of non-negative weak solutions to doubly nonlinear parabolic partial differential equations of the type (Formula presented.) with q>0, ΩT:=Ω×(0,T)⊂Rn+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 Hölder continuity of the spatial gradient of non-negative weak solutions in the super-critical fast diffusion regime (Formula presented.) 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.