Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and $g:\bar \Omega \to \mathbb{R}$ be a continuous (non-constant) function.
Where can I find a proof of the fact that if $u$ solves \begin{equation} \begin{cases} u_t + \Delta u = 0 &\quad \text{ on } (0,\infty) \times \Omega \\ u = 0 &\quad \text{ on } (0,\infty) \times \partial \Omega \\ u(0,\cdot) =g &\quad \text{ on } \bar \Omega \end{cases} \end{equation}
then $$\min_{\bar \Omega} g < u(t,x) <\max_{\bar \Omega} g$$ for any $t>0$ and $x \in \bar \Omega$?
This is the maximum principle for the heat equation which says solutions to the heat equation attains its maximum and minimum on the boundary of $\Omega$ or at time $t=0$. Every PDE book worth its salt should cover this property in its discussion of the heat equation. Here is a proof in a single spatial variable.