Asymptotics for the blow-up boundary solution of the logistic equation with absorption

Cîrstea, Florica-Corina; Rădulescu, Vicenţiu

doi:10.1016/S1631-073X(03)00027-X

Partial Differential Equations

Asymptotics for the blow-up boundary solution of the logistic equation with absorption
[Comportement asymptotique de la solution explosant au bord de l'équation logistique avec absorption]

Présenté par : Philippe G. Ciarlet

Cîrstea, Florica-Corina ¹ ; Rădulescu, Vicenţiu ²

¹ School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia
² Department of Mathematics, University of Craiova, 1100 Craiova, Romania

Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 231-236.

Résumé
Abstract

Soit $Ω$ un domaine borné et régulier de $R^{N}$ . On suppose que f∈C¹[0,∞) est ⩾0 et telle que f(u)/u soit strictement croissante sur (0,+∞). Soit a un réel et b⩾0, b≢0, une fonction continue sur $\bar{Ω}$ telle que b≡0 sur $\partial Ω$ . Dans cette Note on établit le comportement asymptotique de l'unique solution positive du problème logistique Δu+au=b(x)f(u) sur $Ω$ avec la donnée au bord singulière u(x)→+∞ si $dist (x, \partial Ω) \to 0$ . Notre analyse porte sur la théorie de la variation régulière de Karamata.

Let $Ω$ be a smooth bounded domain in $R^{N}$ . Assume that f⩾0 is a C¹-function on [0,∞) such that f(u)/u is increasing on (0,+∞). Let a be a real number and let b⩾0, b≢0 be a continuous function such that b≡0 on $\partial Ω$ . The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δu+au=b(x)f(u) in $Ω$ , subject to the singular boundary condition u(x)→+∞ as $dist (x, \partial Ω) \to 0$ . Our analysis is based on the Karamata regular variation theory.

Reçu le : 2002-12-02
Accepté le : 2002-12-09
Publié le : 2003-02-01

DOI : 10.1016/S1631-073X(03)00027-X

Affiliations des auteurs :

Cîrstea, Florica-Corina ¹ ; Rădulescu, Vicenţiu ²

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Cîrstea, Florica-Corina; Rădulescu, Vicenţiu. Asymptotics for the blow-up boundary solution of the logistic equation with absorption. Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 231-236. doi : 10.1016/S1631-073X(03)00027-X. http://www.numdam.org/articles/10.1016/S1631-073X(03)00027-X/

Bibliographie
Cité par

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[2] F. Cı̂rstea, V. Rădulescu, Solutions with boundary blow-up for a class of nonlinear elliptic problems, Houston J. Math., in press

[3] F. Cı̂rstea, V. Rădulescu, Blow-up solutions of logistic equations with absorption: uniqueness and asymptotics, in preparation

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