Definition of parabolic boundary by Lieberman I have a question concerning the definition of parabolic boundary in the book "Second Order Parabolic Differential Equations" (2nd edition, 1996) by Gary M. Lieberman. He uses the following notation:
$$|X|=\max(|x|,|t|^{1/2}),\quad Q(X,r)=\{Y\in \mathbb{R}^{n+1}\colon |Y-X|<r,s<t\},$$
where $X=(x,t),Y=(y,s)$.
He defines on page 7 for an arbitrary bounded domain $\Omega\subset \mathbb{R}^{n+1}$ the parabolic boundary as the points $X\in \partial \Omega$, where $\partial \Omega$ denotes the topological boundary, such that for any $\epsilon >0$ the cylinder $Q(X,\epsilon)$ contains points which do not belong to $\Omega$.
Now he claims that in the special case $\Omega=D\times (0,T)$, where $D$ is a bounded domain in $\mathbb{R}^n$, the parabolic boundary consists of $B\Omega=D\times \{t=0\}$ (bottom), $C\Omega=\partial D\times \{t=0\}$ (corner), $S\Omega=\partial D\times (0,T)$ (side). Could anyone explain to me why he does not include the set $\partial D\times \{t=T\}$?
 A: Given that the domain is $\Omega = D \times (0,T)$ and the definition of the parabolic boundary of this domain is $\mathcal{P}(\Omega) = \{ X \in \partial \Omega \ | \  \forall \ \epsilon > 0, \ Q(X,\epsilon) \not\subset \Omega \} $:
$\mathcal{P}(\Omega)$ is a  backwards-in-time cylinder anchored at the point $X = (x,t)$. It extends from $t$ to $(t-\epsilon)$.

*

*Using this cylinder on the lower face $B \Omega = D \times \{ t = 0
   \}$ and the boundary of this face $C\Omega = \partial D \times \{ t=0
   \}$ with $\epsilon > 0$, we see no intersection with $\Omega$. Thus,
this is a part of $\mathcal{P}(\Omega)$.


*At $S \Omega = \partial D \times (0,T) $, the cylinder has a partial
intersection with the domain, but there are some points outside the
domain. Thus, this is a part of $\mathcal{P}(\Omega)$.


*For the domain $(D \times {t=T})$, any cylinder anchored on this face
that has a finite $\epsilon$ extends into the domain $\Omega$. Thus,
this face does not belong to the parabolic boundary $\mathcal{P}(\Omega)$.
I hope the schematic helps.$D \times (0,T)$" />
