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.

Abstract (VO)
Résumé

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.

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.

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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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

[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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Zhijun Zhang; Jianning Yu The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem, Journal of Mathematical Analysis and Applications, Volume 329 (2007) no. 2, pp. 1330-1342 | DOI:10.1016/j.jmaa.2006.07.052 | Zbl:1154.35356
Vicenţiu D. Rădulescu Singular Phenomena in Nonlinear Elliptic Problems: From Blow-Up Boundary Solutions to Equations with Singular Nonlinearities, Stationary Partial Differential Equations, Volume 4 (2007), p. 485 | DOI:10.1016/s1874-5733(07)80010-6
Zhijun Zhang; Jianning Yu The exact asymptotic behaviour of the unique solution to a singular Dirichlet problem, Boundary Value Problems, Volume 2006 (2006), p. 10 (Id/No 75674) | DOI:10.1155/bvp/2006/75674 | Zbl:1136.35388
Julián López-Gómez Optimal uniqueness theorems and exact blow-up rates of large solutions, Journal of Differential Equations, Volume 224 (2006) no. 2, pp. 385-439 | DOI:10.1016/j.jde.2005.08.008 | Zbl:1208.35036
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Zhijun Zhang The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems, Journal of Mathematical Analysis and Applications, Volume 308 (2005) no. 2, pp. 532-540 | DOI:10.1016/j.jmaa.2004.11.029 | Zbl:1160.35417
Zhijun Zhang The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation, Journal of Mathematical Analysis and Applications, Volume 312 (2005) no. 1, pp. 33-43 | DOI:10.1016/j.jmaa.2005.03.023 | Zbl:1165.35377
Zhijun Zhang The asymptotic behaviour of solutions with boundary blow-up for semilinear elliptic equations with nonlinear gradient terms, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 62 (2005) no. 6, pp. 1137-1148 | DOI:10.1016/j.na.2005.04.028 | Zbl:1213.35225
Julián López-Gómez Metasolutions: Malthus versus Verhulst in Population Dynamics. A Dream of Volterra, Stationary Partial Differential Equations, Volume 2 (2005), p. 211 | DOI:10.1016/s1874-5733(05)80012-9
Julián López-Gómez The boundary blow-up rate of large solutions., Journal of Differential Equations, Volume 195 (2003) no. 1, pp. 25-45 | DOI:10.1016/j.jde.2003.06.003 | Zbl:1130.35329

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