Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions.

For all $f\in L^2(\Omega)$, we denote by $S(t)f$ the solution of the equation $$ dy/dt=\Delta y,\; y(0)=f. $$

We say that $f\ge 0$ iff $f(x)\ge 0,\; \forall x\in \Omega$.

I have two questions :

1. It follows from the maximum principle that

$ f\ge 0 \;$ implies $\; S(t)f\ge 0,\; \forall t\ge 0$.

Suppose now that $\; S(t_1)f\ge 0,\;$ for some $t_1>0$. Do we have $f\ge0?$

2. Let $f, g \in L^2(\Omega)$ such that $fg\ge0$. Do we have $(S(t)f)(S(t)g)\ge 0,\;\forall t\ge 0?$