Boundary behavior of solutions to the parabolic p-Laplace equation

TL;DR

This paper develops boundary estimates and Harnack principles for non-negative solutions to the degenerate p-parabolic equation in smooth domains, revealing intrinsic waiting time phenomena affecting boundary behavior.

Contribution

It introduces new parabolic intrinsic Harnack chains and boundary decay estimates for the p-parabolic equation in regular domains, advancing the understanding of boundary behavior.

Findings

Established boundary decay estimates for solutions.

Proved boundary Harnack principles in smooth domains.

Identified intrinsic waiting time phenomena affecting boundary solutions.

Abstract

We establish boundary estimates for non-negative solutions to the p-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and H\"older continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.