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

Francesco Petitta

doi:10.1016/j.crma.2007.03.021

Partial Differential Equations

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

Presented by: Haïm Brezis

Francesco Petitta ¹

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

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

Abstract
Résumé

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.

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.

Accepted: 2007-03-18
Published online: 2007-04-30

DOI: 10.1016/j.crma.2007.03.021

Author's affiliations:

Francesco Petitta ¹

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

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     author = {Francesco Petitta},
     title = {Asymptotic behavior of solutions for linear parabolic equations with general measure data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {571--576},
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     volume = {344},
     number = {9},
     year = {2007},
     doi = {10.1016/j.crma.2007.03.021},
     language = {en},
}

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Francesco Petitta. Asymptotic behavior of solutions for linear parabolic equations with general measure data. Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 571-576. doi : 10.1016/j.crma.2007.03.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.021/

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