Boundary behavior of solutions to the parabolic p-Laplace equation II

TL;DR

This paper investigates the short-time boundary behavior of solutions to the p-parabolic equation, revealing a decay-rate dichotomy and its connection to boundary measures, advancing understanding of solution continuation in complex domains.

Contribution

It introduces a novel short-time analysis of boundary solutions and links decay behavior to boundary measures, extending previous long-time focus results.

Findings

Established a decay-rate dichotomy for solutions near the boundary.

Connected decay behavior to the support of boundary Riesz measures.

Implications for solution continuation in NTA-domains.

Abstract

This paper is the second installment in a series of papers concerning the boundary behavior of solutions to the $p$-parabolic equations. In this paper we are interested in the short time behavior of the solutions, which is in contrast with much of the literature, where all results require a waiting time. We prove a dichotomy about the decay-rate of non-negative solutions vanishing on the lateral boundary in a cylindrical $C^{1,1}$ domain. Furthermore we connect this dichotomy to the support of the boundary type Riesz measure related to the $p$-parabolic equation in NTA-domains, which has consequences for the continuation of solutions.