# American Institute of Mathematical Sciences

January & February  2004, 10(1&2): 193-200. doi: 10.3934/dcds.2004.10.193

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

 1 Dpto. Matemática Aplicada, Univ. Complutense, 28040 Madrid, Spain 2 CEREMATH – UMR MIP, Université Toulouse 1, Place A.France, F–31042 Toulouse Cedex, France

Received  March 2003 Revised  August 2003 Published  October 2003

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 $u(t,x)>0 \forall t>t_p, a. e. x\in\Omega$.

Citation: Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193
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