Comptes Rendus
no. 3
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 Cîrstea1 ; Vicenţiu Rădulescu2
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.

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 Ω ¯ telle que b≡0 sur ∂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,∂Ω)→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 ∂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,∂Ω)→0. Our analysis is based on the Karamata regular variation theory.

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

[1] F. Cı̂rstea; V. Rădulescu Uniqueness of the blow-up boundary solution of logistic equations with absorption, C. R. Acad. Sci. Paris, Sér. I, Volume 335 (2002), pp. 447-452

[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

[4] J. García-Melián; R. Letelier-Albornez; J. Sabina de Lis Uniqueness and asymptotic behavior for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., Volume 129 (2001), pp. 3593-3602

[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

[8] E. Seneta Regularly Varying Functions, Lecture Notes in Math., 508, Springer-Verlag, Berlin, 1976

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