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

Florica-Corina Cîrstea; Vicenţiu Rădulescu

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

Florica-Corina Cîrstea ¹ ; Vicenţiu Rădulescu 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, Volume 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 :

Florica-Corina Cîrstea ¹ ; Vicenţiu Rădulescu 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

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     title = {Asymptotics for the blow-up boundary solution of the logistic equation with absorption},
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Florica-Corina Cîrstea; Vicenţiu Rădulescu. Asymptotics for the blow-up boundary solution of the logistic equation with absorption. Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 231-236. doi : 10.1016/S1631-073X(03)00027-X. https://comptes-rendus.academie-sciences.fr/mathematique/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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[5] J. Karamata Sur un mode de croissance régulière de fonctions. Théorèmes fondamentaux, Bull. Soc. Math. France, Volume 61 (1933), pp. 55-62

[6] J.B. Keller On solution of Δu=f(u), Comm. Pure Appl. Math., Volume 10 (1957), pp. 503-510

[7] R. Osserman On the inequality Δu⩾f(u), Pacific J. Math., Volume 7 (1957), pp. 1641-1647

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