Let $\Omega$ a bounded domain in $\mathbb{R}^{n}$ and $u \in C(\bar{\Omega})$. Suposse $u \in C^{2,1}(\Omega \times (0,\infty)))\cap C(\bar{\Omega}\times [0,\infty))$ is a solution of
$u_{t}-\Delta u=0 \ in \ \Omega \times (0,\infty)$
$u(.,0)=u_{0}$ on $\Omega$
$u=0$ on $\partial \Omega \times (0,\infty)$
Prove that
$\sup\limits_{\Omega} |u(,t)|\leq Ce^{-\mu t}\sup\limits_{\Omega}|u_{0}|$ for any $t>0$
where $\mu$ and $C$ are positive constants depending only on $n$ and $\Omega$.
My strategy for this problem is find a integral representation of solution of this problem as well as the case of the heat equation on $\mathbb{R}^{n}\times (0,\infty)$, an idea would be to extend the function $u$ properly, but i don't know how to do this. Any tips?