Capacitary representation of positive solutions of semilinear parabolic equations

Moshe Marcus; Laurent Véron

doi:10.1016/j.crma.2006.02.033

Partial Differential Equations

Capacitary representation of positive solutions of semilinear parabolic equations
[Représentation capacitaire des solutions positives d'équations paraboliques semi-linéaires]

Présenté par : Haïm Brezis

Moshe Marcus ¹ ; Laurent Véron ²

¹ Department of Mathematics, Technion, Haifa 32000, Israel
² Laboratoire de mathématiques, faculté des sciences, parc de Grandmont, 37200 Tours, France

Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 655-660.

Résumé
Abstract

Nous donnons une estimation bilatérale précise de la solution maximale ${\bar{u}}_{F}$ de $\partial_{t} u - Δ u + u^{q} = 0$ dans $R^{N} \times (0, \infty)$ , $q > 1$ , $N ⩾ 1$ , qui s'annulle en $t = 0$ sur le complémentaire d'un sous-ensemble fermé $F \subset R^{N}$ . Cette estimation s'exprime par un test de Wiener impliquant la capacité de Bessel $C_{2 / q, q^{'}}$ . Nous déduisons de cette estimation que ${\bar{u}}_{F}$ est σ-moderée au sens de Dynkin.

We give a global bilateral estimate on the maximal solution ${\bar{u}}_{F}$ of $\partial_{t} u - Δ u + u^{q} = 0$ in $R^{N} \times (0, \infty)$ , $q > 1$ , $N ⩾ 1$ , which vanishes at $t = 0$ on the complement of a closed subset $F \subset R^{N}$ . This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2 / q, q^{'}}$ . We deduce from this estimate that ${\bar{u}}_{F}$ is σ-moderate in Dynkin's sense.

Reçu le : 2006-02-22
Publié le : 2006-03-27

DOI : 10.1016/j.crma.2006.02.033

Affiliations des auteurs :

Moshe Marcus ¹ ; Laurent Véron ²

¹ Department of Mathematics, Technion, Haifa 32000, Israel
² Laboratoire de mathématiques, faculté des sciences, parc de Grandmont, 37200 Tours, France

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     author = {Moshe Marcus and Laurent V\'eron},
     title = {Capacitary representation of positive solutions of semilinear parabolic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {655--660},
     publisher = {Elsevier},
     volume = {342},
     number = {9},
     year = {2006},
     doi = {10.1016/j.crma.2006.02.033},
     language = {en},
}

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Moshe Marcus; Laurent Véron. Capacitary representation of positive solutions of semilinear parabolic equations. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 655-660. doi : 10.1016/j.crma.2006.02.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.033/

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