Capacitary representation of positive solutions of semilinear parabolic equations

Marcus, Moshe; Véron, Laurent

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

Marcus, Moshe ¹ ; Véron, Laurent 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, Tome 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 :

Marcus, Moshe ¹ ; Véron, Laurent 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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     title = {Capacitary representation of positive solutions of semilinear parabolic equations},
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Marcus, Moshe; Véron, Laurent. Capacitary representation of positive solutions of semilinear parabolic equations. Comptes Rendus. Mathématique, Tome 342 (2006) no. 9, pp. 655-660. doi : 10.1016/j.crma.2006.02.033. http://www.numdam.org/articles/10.1016/j.crma.2006.02.033/

Bibliographie
Cité par

[1] Adams, D.R.; Hedberg, L.I. Function Spaces and Potential Theory, Grundlehren Math. Wiss., vol. 314, Springer, 1996

[2] Aikawa, H.; Borichev, A.A. Quasiadditivity and measure property of capacity and the tangential boundary behaviour of harmonic functions, Trans. Amer. Math. Soc., Volume 348 (1996), pp. 1013-1030

[3] Baras, P.; Pierre, M. Problèmes paraboliques semi-linéaires avec données mesures, Appl. Anal., Volume 18 (1984), pp. 111-149

[4] Brezis, H.; Friedman, A. Nonlinear parabolic equations involving measures as initial data, J. Math. Pures Appl., Volume 62 (1983), pp. 73-97

[5] Brezis, H.; Peletier, L.A.; Terman, D. A very singular equation of the heat equation with absorption, Arch. Ration. Mech. Anal., Volume 95 (1986), pp. 185-209

[6] Marcus, M.; Véron, L. The initial trace of positive solutions of semilinear parabolic equations, Comm. Partial Differential Equations, Volume 24 (1999), pp. 1445-1499

[7] Marcus, M.; Véron, L. Capacitary estimates of positive solutions of semilinear elliptic equations with absorption, J. Eur. Math. Soc., Volume 6 (2004), pp. 483-527

[8] Triebel, H. Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978

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