I cannot believe I am asking this but here we go. This is from literature related to parabolic equations. Let $D$ be a bounded $n+1$ dimensional domain in $R^{n+1}$, $(x,t)=(x_1,\dots,x_n,t)$ is a variable point in $R^{n+1}$. D is bounded by a domain $B$ on $t=0$ and a domain $B$ on $t=0$, a domain $B_T$ on $t=T$ and a manifold $S$ lying on $0<t\leq T$. $S+\bar{B}$ is the normal boundary of $D$
what exactly is a domain? Is this just a subset of $R^{n+1}$.
The typical application would have $D=S\times(0,T)$. Where $S$ is a circle, say.
However, let's just take $n=1$ for simplicity sake take $D$ to be a closed unit circle with centre at $(1,0)$, with its boundary $C$, then $B=\{((0,0)\}$, $B_T=\{(2,0)\}$ and $S$ is everything not in $C$ and $B_T$.
is this allowed?
The reason I am asking is this is that I am wondering if the standard results for parabolic equation hold for domains of this type. There seems to be nothing forbidding the example I have just given.
Typically in PDE a "domain" refers to an open connected set. To be fair, though, this notation is not in universal use.