Regularity of soda-can domains for the $p$-parabolic equation

Abstract

We classify the regularity of the origin for the $p$-parabolic equation in the soda-can domains $${\Theta }_{l,\theta }=\{(x,t)\in {\mathbb{R}}^n\times \mathbb{R}:0<-t<\theta |x{|}^l<\theta \},\text{ where }l,\theta >0.$$ The classification covers every $p>1$ in dimensions $n\ge 2$. At the endpoint $l=p$, the origin is regular if and only if $p\ge 2n/(n+1)$. We also prove that the origin admits a traditional barrier for every $p>2$ and $l>0$. When $2<p<n$ and $l<p$, the origin is nevertheless irregular. This answers negatively the well-known open question whether one barrier suffices to characterize boundary regularity in the degenerate range $p>2$.