Let $\Omega$ be a bounded domain in $\mathbb{R^n}$ and $u_0\in C(\bar{\Omega})$ and if $u\in C^{2,1}(\Omega\times (0,\infty))\cap C((\bar{\Omega})×[0,\infty))$ is a solution of $ u_t-\Delta u=0$ in $\Omega\times (0,\infty)$, $u(.,0)=u_0$ on $\Omega$ and $u=0$ on $\partial\Omega \times (0,\infty)$ then to show that
$ sup_{\Omega}|u(.,t)|\leq Ce^{-\mu t} sup_{\Omega}|u_0|$ for every $t>0$.
I am not getting which function to start with to apply the maximum principle to get the desired estimate.