Extremal singular solutions for degenerate logistic-type equations in anisotropic media

Florica-Corina Cîrstea; Vicenţiu Rădulescu

doi:10.1016/j.crma.2004.04.025

Partial Differential Equations

Extremal singular solutions for degenerate logistic-type equations in anisotropic media
[Solutions singulières extremales des équations du type logistique en milieu anisotrope.]

Présenté par : Haïm Brezis

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 University of Craiova, Department of Mathematics, 13 A. I. Cuza Street, 200585 Craiova, Romania

Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 119-124.

Résumé
Abstract

Soit Ω un domaine borné et régulier de $R^{N}$ . Soit $b ⩾ 0$ , $b \equiv̸ 0$ une fonction continue dans $\bar{Ω}$ et $D_{0} \neq \emptyset$ un sous-ensemble fermé de $[b = 0]$ . On étudie le problème logistique $Δ u + a u = b (x) f (u)$ dans $Ω ∖ D_{0}$ , $B u = 0$ sur ∂Ω, et $u = + \infty$ sur $\partial D_{0}$ , où a est un réel, $B$ désigne ou bien une condition de Dirichlet ou bien une condition mixte sur ∂Ω, et $f ⩾ 0$ est une fonction régulière telle que l'application $f (u) / u$ soit croissante sur $(0, \infty)$ . Dans cette Note on établit l'existence des solutions singulières extremales, un résultat d'unicité et on décrit également la vitesse d'explosion au bord.

Let Ω be a smooth bounded domain in $R^{N}$ . Let $b ⩾ 0$ , $b \equiv̸ 0$ be a continuous function on $\bar{Ω}$ and consider a closed subset $D_{0} \neq \emptyset$ of $[b = 0]$ . We study the logistic problem $Δ u + a u = b (x) f (u)$ in $Ω ∖ D_{0}$ , $B u = 0$ on ∂Ω, and $u = + \infty$ on $\partial D_{0}$ , where a is a real number, $B$ denotes either the Dirichlet or the mixed boundary operator, and $f ⩾ 0$ is a smooth function such that $f (u) / u$ is increasing on $(0, \infty)$ . In this Note we establish the existence of extremal singular solutions to the above problem, a uniqueness result, and we describe the blow-up at the boundary.

Reçu le : 2004-04-01
Publié le : 2004-07-15

DOI : 10.1016/j.crma.2004.04.025

Affiliations des auteurs :

Florica-Corina Cîrstea ¹ ; Vicenţiu Rădulescu ²

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     author = {Florica-Corina C{\^\i}rstea and Vicen\c{t}iu R\u{a}dulescu},
     title = {Extremal singular solutions for degenerate logistic-type equations in anisotropic media},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {119--124},
     publisher = {Elsevier},
     volume = {339},
     number = {2},
     year = {2004},
     doi = {10.1016/j.crma.2004.04.025},
     language = {en},
}

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Florica-Corina Cîrstea; Vicenţiu Rădulescu. Extremal singular solutions for degenerate logistic-type equations in anisotropic media. Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 119-124. doi : 10.1016/j.crma.2004.04.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.025/

Bibliographie
Cité par

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

[2] F.-C. Cîrstea; 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

[3] F.-C. Cîrstea, V. Rădulescu, Nonlinear problems with singular boundary conditions arising in population dynamics: a Karamata regular variation theory approach, submitted for publication

[4] 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

[5] 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

[6] S.I. Resnick Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, Berlin, 1987

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

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