Asymptotic behavior of solutions for linear parabolic equations with general measure data

Petitta, Francesco

doi:10.1016/j.crma.2007.03.021

Partial Differential Equations

Asymptotic behavior of solutions for linear parabolic equations with general measure data
[Comportement asymptotique des solutions des équations paraboliques linéaires avec données de mesures générales]

Présenté par : Haïm Brezis

Petitta, Francesco ¹

¹ Dipartimento di Matematica, Università La Sapienza, Piazzale A. Moro, 2, 00185 Roma, Italy

Comptes Rendus. Mathématique, Tome 344 (2007) no. 9, pp. 571-576.

Résumé
Abstract

Dans cette Note nous traitons le comportement asymptotique, quand t tend vers l'infini, des solutions des équations paraboliques linéaires dont le modéle est :

{\begin{matrix} u_{t} - Δ u = μ & dans (0, T) \times Ω, \\ u (0, x) = u_{0} & dans Ω, \end{matrix}

oú μ est une mesure de Radon générale, éventuellement singulière, qui ne dépend pas de t, et où

u_{0} \in L^{1} (Ω)

. Nous montrons que la solution de dualité, qui existe et est unique, converge vers la solution de dualité (introduite par Stampacchia (1965)) du probléme elliptique associé.

In this Note we deal with the asymptotic behavior as t tends to infinity of solutions for linear parabolic equations whose model is

{\begin{matrix} u_{t} - Δ u = μ & in (0, T) \times Ω, \\ u (0, x) = u_{0} & in Ω, \end{matrix}

where μ is a general, possibly singular, Radon measure which does not depend on time, and

u_{0} \in L^{1} (Ω)

. We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by Stampacchia (1965)) of the associated elliptic problem.

Accepté le : 2007-03-18
Publié le : 2007-04-30

DOI : 10.1016/j.crma.2007.03.021

Affiliations des auteurs :

Petitta, Francesco ¹

¹ Dipartimento di Matematica, Università La Sapienza, Piazzale A. Moro, 2, 00185 Roma, Italy

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Petitta, Francesco. Asymptotic behavior of solutions for linear parabolic equations with general measure data. Comptes Rendus. Mathématique, Tome 344 (2007) no. 9, pp. 571-576. doi : 10.1016/j.crma.2007.03.021. http://www.numdam.org/articles/10.1016/j.crma.2007.03.021/

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