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

Soit Ω un domaine borné et régulier de R N . On suppose que fC1[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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00027-X
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
@article{CRMATH_2003__336_3_231_0,
     author = {C{\^\i}rstea, Florica-Corina and R\u{a}dulescu, Vicen\c{t}iu},
     title = {Asymptotics for the blow-up boundary solution of the logistic equation with absorption},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {231--236},
     publisher = {Elsevier},
     volume = {336},
     number = {3},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00027-X},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(03)00027-X/}
}
TY  - JOUR
AU  - Cîrstea, Florica-Corina
AU  - Rădulescu, Vicenţiu
TI  - Asymptotics for the blow-up boundary solution of the logistic equation with absorption
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 231
EP  - 236
VL  - 336
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(03)00027-X/
DO  - 10.1016/S1631-073X(03)00027-X
LA  - en
ID  - CRMATH_2003__336_3_231_0
ER  - 
%0 Journal Article
%A Cîrstea, Florica-Corina
%A Rădulescu, Vicenţiu
%T Asymptotics for the blow-up boundary solution of the logistic equation with absorption
%J Comptes Rendus. Mathématique
%D 2003
%P 231-236
%V 336
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(03)00027-X/
%R 10.1016/S1631-073X(03)00027-X
%G en
%F CRMATH_2003__336_3_231_0
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://archive.numdam.org/articles/10.1016/S1631-073X(03)00027-X/

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

[5] Karamata, J. Sur un mode de croissance régulière de fonctions. Théorèmes fondamentaux, Bull. Soc. Math. France, Volume 61 (1933), pp. 55-62

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

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

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

Cité par Sources :