Continuity estimates for doubly degenerate parabolic equations with lower order terms via nonlinear potentials

Abstract

This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. A model equation is \begin{equation*}\partial_t u-\operatorname{div}(u^{m-1}|Du|^{p-2}Du)=f\qquad \text{in}\quad\Omega\times(-T,0)\subset \mathbb{R}^{n+1}.\end{equation*} Here, we consider the case $m>1$ and $2<p<n$. We establish a continuity estimate for $u$ in terms of elliptic Riesz potentials of the right-hand side of the equation.