Uniqueness of the blow-up boundary solution of logistic equations with absorbtion

Florica-Corina Cı̂rstea; Vicenţiu Rădulescu

doi:10.1016/S1631-073X(02)02503-7

Uniqueness of the blow-up boundary solution of logistic equations with absorbtion
[Unicité de la solution explosant au bord pour équations logistiques avec absorption]

Florica-Corina Cı̂rstea ¹ ; Vicenţiu Rădulescu ²

¹ School of Communications and Informatics, Victoria University of Technology, P.O. Box 14428, Melbourne City MC, Victoria 8001, Australia
² Department of Mathematics, University of Craiova, 1100 Craiova, Romania

Comptes Rendus. Mathématique, Volume 335 (2002) no. 5, pp. 447-452.

Résumé
Abstract

Soit $Ω$ un domaine borné et régulier de $ℝ^{N}$ . On suppose que f∈C¹[0,∞) est une fonction non-negative telle que f(u)/u soit strictement croissante sur (0,+∞). Soit a un réel et b⩾0, $b / \equiv 0$ une fonction continue sur $\bar{Ω}$ . On étudie l'équation logistique Δu+au=b(x)f(u) sur $Ω$ . Le but de cette Note est de montrer l'unicité de la solution explosant au bord de $Ω$ dans un contexte général, qui apparaı̂t en théorie des probabilités.

Let $Ω$ be a smooth bounded domain in $ℝ^{N}$ . Assume f∈C¹[0,∞) is a non-negative function such that f(u)/u is increasing on (0,∞). Let a be a real number and let b⩾0, $b / \equiv 0$ be a continuous function such that b≡0 on $\partial Ω$ . We study the logistic equation Δu+au=b(x)f(u) in $Ω$ . The special feature of this work is the uniqueness of positive solutions blowing-up on $\partial Ω$ , in a general setting that arises in probability theory.

Reçu le : 2002-07-01
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02503-7

Affiliations des auteurs :

Florica-Corina Cı̂rstea ¹ ; Vicenţiu Rădulescu ²

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Florica-Corina Cı̂rstea; Vicenţiu Rădulescu. Uniqueness of the blow-up boundary solution of logistic equations with absorbtion. Comptes Rendus. Mathématique, Volume 335 (2002) no. 5, pp. 447-452. doi : 10.1016/S1631-073X(02)02503-7. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02503-7/

Bibliographie
Cité par

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

[4] F. Cı̂rstea; V. Rădulescu Existence and uniqueness of blow-up solutions for a class of logistic equations, Commun. Contemp. Math., Volume 4 (2002), pp. 559-586

[5] Y. Du; Q. Huang Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., Volume 31 (1999), pp. 1-18

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

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[8] C. Loewner; L. Nirenberg Partial differential equations invariant under conformal or projective transformations (L. Alhfors, ed.), Contributions to Analysis, Academic Press, New York, 1974, pp. 245-272

[9] M. Marcus; L. Véron Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 14 (1997), pp. 237-274

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

[12] L. Véron Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., Volume 59 (1992), pp. 231-250

Cité par Sources :

^☆ The research of F. Cı̂rstea was done under the IPRS Programme funded by the Australian Government through DETYA. V. Rădulescu was supported by the P.I.C.S. Research Programme between France and Romania.

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