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 \begin{align*} \partial_t u^q - \mbox{div}\, A(x,t,Du)=0 \qquad\mbox{in $\hat{I}{\text{c◯}}_T$}, \end{align*} with $q>0$, $\hat{I}{\text{c◯}}_T=\hat{I}\text{c◯}\times (0,T)\subset {\mathbb{R}}^{n+1}$ a space-time cylinder, and $A=A(x,t,\hat{I}¾)$ 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 $$0<p-1<q<\frac{n(p-1)}{(n-p{)}_{+}}.$$ 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 $L^{\infty }$-bound for the spatial gradient.