Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle

Abstract

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$.