Barriers, Barenblatt solutions and regularity of soda can domains for the heat equation and nonlinear $p$-parabolic equations

TL;DR

This paper characterizes the regularity of the origin as a boundary point for soda can domains in the heat and nonlinear p-parabolic equations, extending classical results to nonconvex and symmetric convex cases.

Contribution

It provides a complete characterization of boundary regularity for soda can domains for certain p-values, generalizing Petrovskii's classical results to nonlinear equations.

Findings

Complete regularity criteria for p<2n/(n+1) and p=2.

Extension of Petrovskii's classical results to nonlinear p-parabolic equations.

Analysis of domains with nonconvex and symmetric convex time sections.

Abstract

In this paper we study when the origin $(0,0)$ is a regular (or irregular) boundary point for the so-called soda can domains of the type \[ \Theta_{l,\theta}:= \{(x,t) \in \mathbf{R}^{n+1}: 0<-t < \theta |x|^l <\theta\}, \quad \text{with $l,\theta >0$,} \] for the $p$-parabolic equation $\partial_t u- \Delta_p u=0$, where $1<p<\infty$. For $p<2n/(n+1)$ and for the heat equation (i.e.\ $p=2$) we completely determine when the origin is regular for soda can domains. The domains $\Theta_{l,\theta}$ have nonconvex time sections with power dependence on time. For domains with rotationally symmetric convex time sections with power dependence on time, the regularity of the origin as the last point was characterized by Petrovskii (in 1935) for the heat equation, and almost completely in the nonlinear case ($p \ne 2$) in our earlier paper (joint with Gianazza, Math. Ann. 368 (2017), 885--904).