Monotonicity in integrodifferential equations

Jérôme Coville

doi:10.1016/j.crma.2003.07.005

Partial Differential Equations

Monotonicity in integrodifferential equations

Presented by: Haïm Brezis

Jérôme Coville ¹

¹ Laboratoire Jacques Louis Lions, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 445-450.

Abstract (Original language)
Résumé

We study the behavior of positive solutions of the Dirichlet problem Lu=f(u) in $Ω$ with $Ω = (a, + \infty)$ , where a can be −∞, and L is an abstract operator which is non-increasing under translation and satisfies a strong maximum principle property. This covers the case of many integral operators. Under some assumptions on f (e.g., bistable, monostable), we show that any solution exhibits a monotone behavior.

On présente plusieurs résultats concernant le comportement des solutions positives du problème de Dirichlet Lu=f(u) sur un ouvert $Ω$ , où $Ω = (a, + \infty)$ avec a pouvant être égale à −∞. Ici, L est un opérateur vérifiant un principe du maximum fort ainsi qu'une propriété de décroissance par translation. Nos résultats couvrent le cas d'opérateurs intégraux. On établit le caractère monotone des solutions pour certaines classes de nonlinéarités f.

Accepted: 2003-07-17
Published online: 2003-09-26

DOI: 10.1016/j.crma.2003.07.005

Author's affiliations:

Jérôme Coville ¹

¹ Laboratoire Jacques Louis Lions, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France

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Jérôme Coville. Monotonicity in integrodifferential equations. Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 445-450. doi : 10.1016/j.crma.2003.07.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.07.005/

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