Maximum principle for the heat equation with Dirichlet conditions

Question

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

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

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

Answer 1

As TrialAndError said, the answer to (1) is negative. You can get an (sort of) explicit counterexample by combining two fundamental solutions with nearby sources; say, $2G(x,t;w)-G(x,t;w')$. For any fixed $t_1>0$ the solution converges to $G(x,t_1,w)$ as $w'\to w$; in particular it is positive when $|w'-w|$ is small enough.

A counterexample to (2) is easier to obtain. Take two continuous functions, $f\ge 0$ and $g\le 0$; with disjoint nonempty supports. Then $fg\equiv 0$ but for every $t>0$ we have $S(t)f>0$ everywhere and $S(t)g<0$ everywhere.