Let $M \subset \mathbb R^3$ compact with no boundary and assume that
$\begin{align} \partial_t u-\Delta u&>0 \quad \text{in} \;M\times(0,T]\\u(\cdot,0) &\geq 0\quad \text{in} \;M \end{align}$
In my opinion, a weak maximum principle implies that $u \geq 0\;.$ However, my professor claims that we deduce a strictly positive $u$ using the strong maximum principle. Why is this true? I can't see why the strong maximum principle could be applied here in any case.
Any help is much appreciated!