For the heat equation, why does the strong maximum principle imply infinite propagation speed of disturbances for bounded domains?

Question

Specifically, let $u_0$ be some continuous initial condition with compact support which is non-zero at some point in the bounded open set $U \subset \mathbb{R}^n$ and let us consider zero Dirichlet conditions on $\partial U$. Why is it true that $u(x,t) > 0$ for any $x \in U$ and $t > 0$ and how can this be deduced from the maximum principle?

Answer 1

Nevermind, I figured it out! It follows from the minimum principle. Assume that it's not true then there exists some point $(x_0,t_0)$ in the interior of the parabolic cylinder such that $\displaystyle u(x_0,t_0)=0=\min_{\bar{U} \times [0,T]} u$ but then by the strong minimum principle, we get that u(x,t) = 0 for all $(x,t) \in \bar{U} \times [0,T]$ which is a contradiction since the initial condition must be positive somewhere.