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

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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

Jesus Ildefonso Díaz ^1, and Jacqueline Fleckinger-Pellé ^2,

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

Abstract
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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$.

Keywords: parabolic problems., heat equation, Maximum and antimaximum principle.

Mathematics Subject Classification: 35B30, 35B50, 35K2.

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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