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 $$Θ_{l,θ}={(x,t)\in\mathbb{R}^{n\times\mathbb{R}:0} &lt; -t &lt; θ|x|^l &lt; θ},\text{ where }l,θ&gt;0.$$ The classification covers every $p&gt;1$ in dimensions $n\ge2$. At the endpoint $l=p$, the origin is regular if and only if $p\ge2n/(n+1)$. We also prove that the origin admits a traditional barrier for every $p&gt;2$ and $l&gt;0$. When $2 < p < n$ and $l &lt; p$, 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 $p&gt;2$.