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

TL;DR

This paper establishes continuity estimates for solutions to inhomogeneous doubly nonlinear parabolic equations with lower order terms, using nonlinear potentials and Riesz potentials to relate solution regularity to the right-hand side data.

Contribution

It introduces a novel continuity estimate for solutions of doubly degenerate parabolic equations based on nonlinear potential theory, extending previous regularity results.

Findings

Continuity of solutions is characterized via nonlinear potentials.

The estimates depend on elliptic Riesz potentials of the inhomogeneous term.

Results apply to equations with parameters m>1 and 2<p<n.

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.