Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 193 - 200, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.193

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Jesus Ildefonso Díaz - Dpto. Matemática Aplicada, Univ. Complutense, 28040 Madrid, Spain (email)
Jacqueline Fleckinger-Pellé - CEREMATH – UMR MIP, Université Toulouse 1, Place A.France, F–31042 Toulouse Cedex, France (email)

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$.

Keywords:  Maximum and antimaximum principle, heat equation, parabolic problems.
Mathematics Subject Classification:  35B30, 35B50, 35K20.

Received: March 2003;      Revised: August 2003;      Available Online: October 2003.