We study the positivity, for large time, of
the solutions to
the heat equation $\mathcal Q_a(f,u^0)$:
$\mathcal Q_a(f,u^0)\qquad$
$\partial_tu-\Delta u=au+f(t,x),$ in $Q=]0,\infty [ \times
\Omega, $
$u(t,x)=0\qquad$ $(t,x)\in ]0,\infty [ \times \partial \Omega,$
$u(0,x)=u^0(x), \qquad x\in \Omega,$
where
$\Omega$ is a smooth bounded domain in $\mathbb R^N$ and
$a\in\mathbb R$.
We obtain some sufficient conditions for having a finite time $t_p>0$
(depending on $a$ and on the data $u^0$ and $f$ which are not necessarily
of the same sign) such that
$ u(t,x)>0 \forall t>t_p, a. e. x\in\Omega$.
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