# What's a space-time cylinder?

What's a space-time cylinder?

As given here:

Proof of weak maximum principle for heat equation

It is the domain

$$Q_T:=\Omega \times (0,T)$$

where $\Omega \subset \mathbb{R}^n$ is the space domain, which shall be bounded. Now it depends on the source whether the time or the space comes first; and whether $0$ and/or $T$ are included in the time domain. For example see Evans "Partial Differential Equations" at the beginning of Chapter 7 where he looks at evolution equations.

EDIT: The notation of cylindric is quite loose, it means $Q_T$ has a kind-of-cylindric shape. $\Omega$ does not have to be a circle or something, it shall just be a bounded domain in $\mathbb{R}^n$. For example in "Partial Differential Equations in Action" by Salso on page 32 the following is an example of a space-time cylinder:

• This is a simple answer. – mavavilj Dec 10 '17 at 18:15
• Although, what's a "cylindric space domain"? You mean the volume of the cylinder? Or $\mathbb{R}^n$ for some $n$? – mavavilj Dec 10 '17 at 22:30
• @mavavilj Please see my edit. I hope it is clearer now. – Fritz Dec 10 '17 at 23:00