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 $(S(t)f)(S(t)g)\ge 0,\;\forall t\ge 0?$



Maximum principle for the heat equation with Dirichlet conditions
The answer to all these questions is "no". Think of $S$ as an averaging operator. Initial temperature can be mostly positive on most of $\Omega$ but somewhere slightly negative. After some time $t\_1$ it will be positive everywhere. This answers the first question. Second one is similar.


