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

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

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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/

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Giovanni Porru; Claudia Anedda Boundary estimates for solutions of weighted semilinear elliptic equations, Discrete and Continuous Dynamical Systems, Volume 32 (2012) no. 11, p. 3801 | DOI:10.3934/dcds.2012.32.3801
Salomón Alarcón; Jorge García-Melián; Alexander Quaas Keller-Osserman type conditions for some elliptic problems with gradient terms, Journal of Differential Equations, Volume 252 (2012) no. 2, pp. 886-914 | DOI:10.1016/j.jde.2011.09.033 | Zbl:1235.35096
Zhijun Zhang; Bo Li The boundary behavior of the unique solution to a singular Dirichlet problem, Journal of Mathematical Analysis and Applications, Volume 391 (2012) no. 1, pp. 278-290 | DOI:10.1016/j.jmaa.2012.02.010 | Zbl:1241.35086
Dušan Repovš Asymptotics for singular solutions of quasilinear elliptic equations with an absorption term, Journal of Mathematical Analysis and Applications, Volume 395 (2012) no. 1, pp. 78-85 | DOI:10.1016/j.jmaa.2012.05.017 | Zbl:1250.35109
Ling Mi; Yanfeng Qi; Bin Liu Blow-up rate of the unique solution for a class of one-dimensional $p$ -Laplacian equations, Nonlinear Analysis. Real World Applications, Volume 13 (2012) no. 6, pp. 2734-2746 | DOI:10.1016/j.nonrwa.2012.03.015 | Zbl:1266.34053
Ling Mi; Bin Liu Second order expansion for blowup solutions of semilinear elliptic problems, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 75 (2012) no. 4, pp. 2591-2613 | DOI:10.1016/j.na.2011.11.002 | Zbl:1253.35057
Huiling Li; Peter Y. H. Pang; Mingxin Wang Boundary blow-up solutions of $p$ -Laplacian elliptic equations with lower order terms, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 63 (2012) no. 2, pp. 295-311 | DOI:10.1007/s00033-011-0175-7 | Zbl:1242.35063
Zong Ming Guo; Yao Yong Yan Transition-layer solutions of quasilinear elliptic boundary blow-up problems and Dirichlet problems, Acta Mathematica Sinica. English Series, Volume 27 (2011) no. 11, pp. 2177-2190 | DOI:10.1007/s10114-011-9327-0 | Zbl:1234.35025
Shuibo Huang; Qiaoyu Tian Boundary behaviour of explosive solution to quasilinear elliptic problems with nonlinear gradient terms, Applicable Analysis, Volume 90 (2011) no. 9, p. 1391 | DOI:10.1080/00036811.2010.524159
Dušan Repovš Singular solutions of perturbed logistic-type equations, Applied Mathematics and Computation, Volume 218 (2011) no. 8, pp. 4414-4422 | DOI:10.1016/j.amc.2011.10.018 | Zbl:1239.35161
Shuibo Huang; Wan-Tong Li; Qiaoyu Tian; Chunlai Mu Large solution to nonlinear elliptic equation with nonlinear gradient terms, Journal of Differential Equations, Volume 251 (2011) no. 12, pp. 3297-3328 | DOI:10.1016/j.jde.2011.08.031 | Zbl:1231.35068
Jorge García-Melián Multiplicity of positive solutions to boundary blow-up elliptic problems with sign-changing weights, Journal of Functional Analysis, Volume 261 (2011) no. 7, pp. 1775-1798 | DOI:10.1016/j.jfa.2011.05.018 | Zbl:1387.35308
Shuibo Huang; Qiaoyu Tian Boundary blow-up rates of large solutions for elliptic equations with convection terms, Journal of Mathematical Analysis and Applications, Volume 373 (2011) no. 1, pp. 30-43 | DOI:10.1016/j.jmaa.2010.06.031 | Zbl:1201.35105
Zhijun Zhang The second expansion of the solution for a singular elliptic boundary value problem, Journal of Mathematical Analysis and Applications, Volume 381 (2011) no. 2, pp. 922-934 | DOI:10.1016/j.jmaa.2011.04.018 | Zbl:1221.35174
Zhijun Zhang The second expansion of large solutions for semilinear elliptic equations, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 74 (2011) no. 11, pp. 3445-3457 | DOI:10.1016/j.na.2011.02.031 | Zbl:1217.35074
Jing Mo; Zuodong Yang Boundary asymptotic behavior and uniqueness of large solutions to quasilinear elliptic equations, Computers Mathematics with Applications, Volume 59 (2010) no. 6, pp. 2007-2017 | DOI:10.1016/j.camwa.2009.12.003 | Zbl:1189.35140
Zhijun Zhang; Yunjie Ma; Ling Mi; Xiaohong Li Blow-up rates of large solutions for elliptic equations, Journal of Differential Equations, Volume 249 (2010) no. 1, pp. 180-199 | DOI:10.1016/j.jde.2010.02.019 | Zbl:1191.35137
Sabrine Gontara; Habib Mâagli; Syrine Masmoudi; Sameh Turki Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, Journal of Mathematical Analysis and Applications, Volume 369 (2010) no. 2, pp. 719-729 | DOI:10.1016/j.jmaa.2010.04.008 | Zbl:1196.35109
Zhijun Zhang Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 73 (2010) no. 10, pp. 3348-3363 | DOI:10.1016/j.na.2010.07.017 | Zbl:1198.35077
Huabing Feng; Chengkui Zhong Boundary behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 73 (2010) no. 10, pp. 3472-3478 | DOI:10.1016/j.na.2009.11.052 | Zbl:1200.35132
Shuibo Huang; Qiaoyu Tian; Shengzhi Zhang; Jinhua Xi; Zheng-En Fan The exact blow-up rates of large solutions for semilinear elliptic equations, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 73 (2010) no. 11, pp. 3489-3501 | DOI:10.1016/j.na.2010.06.069 | Zbl:1200.35134
Ni Xiang Boundary asymptotical behavior of large solutions to complex Hessian equations, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 73 (2010) no. 12, pp. 3940-3946 | DOI:10.1016/j.na.2010.08.027 | Zbl:1215.32018
Yujuan Chen; Mingxin Wang Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 61 (2010) no. 2, pp. 277-292 | DOI:10.1007/s00033-009-0015-1 | Zbl:1273.76395
Shuibo Huang; Qiaoyu Tian; Chunlai Mu Asymptotic behavior of large solution to elliptic equation of Bieberbach-Rademacher type with convection terms, Applied Mathematics and Computation, Volume 210 (2009) no. 2, pp. 284-293 | DOI:10.1016/j.amc.2008.12.067 | Zbl:1178.35167
Shuibo Huang; Qiaoyu Tian Asymptotic behavior of large solution for boundary blowup problems with non-linear gradient terms, Applied Mathematics and Computation, Volume 215 (2009) no. 8, pp. 3091-3097 | DOI:10.1016/j.amc.2009.10.002 | Zbl:1182.35052
Zhifu Xie Uniqueness and blow-up rate of large solutions for elliptic equation $- Δ u = λ u - b (x) h (u)$ , Journal of Differential Equations, Volume 247 (2009) no. 2, pp. 344-363 | DOI:10.1016/j.jde.2009.04.001 | Zbl:1173.35064
S. Cano-Casanova; J. López-Gómez Blow-up rates of radially symmetric large solutions, Journal of Mathematical Analysis and Applications, Volume 352 (2009) no. 1, pp. 166-174 | DOI:10.1016/j.jmaa.2008.06.022 | Zbl:1163.35015
Zhijun Zhang; Yiming Guo; Huabing Feng Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term, Journal of Mathematical Analysis and Applications, Volume 352 (2009) no. 1, pp. 77-84 | DOI:10.1016/j.jmaa.2008.01.075 | Zbl:1163.35022
Peng Feng Remarks on large solutions of a class of semilinear elliptic equations, Journal of Mathematical Analysis and Applications, Volume 356 (2009) no. 2, pp. 393-404 | DOI:10.1016/j.jmaa.2009.03.021 | Zbl:1167.35017
Jorge García-Melián Uniqueness of positive solutions for a boundary blow-up problem, Journal of Mathematical Analysis and Applications, Volume 360 (2009) no. 2, pp. 530-536 | DOI:10.1016/j.jmaa.2009.06.077 | Zbl:1182.35006
Shuibo Huang; Qiaoyu Tian Asymptotic behavior of large solutions to $p$ -Laplacian of Bieberbach-Rademacher type, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 71 (2009) no. 11, pp. 5773-5780 | DOI:10.1016/j.na.2009.04.064 | Zbl:1176.35080
Wei Dong; Juanfei Li; Lishan Liu Uniqueness of boundary blow-up solutions on unbounded domain of $R^{N}$ , Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 71 (2009) no. 12, p. e2118-e2126 | DOI:10.1016/j.na.2009.03.084 | Zbl:1239.35004
Sonia Ben Othman; Habib Mâagli; Syrine Masmoudi; Malek Zribi Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 71 (2009) no. 9, pp. 4137-4150 | DOI:10.1016/j.na.2009.02.073 | Zbl:1177.35091
Zhang Zhijiun The existence and global optimal asymptotic behaviour of large solutions for a semilinear elliptic problem, Acta Mathematica Scientia, Volume 28 (2008) no. 3, p. 595 | DOI:10.1016/s0252-9602(08)60062-4
Zhijun Zhang Asymptotic behavior of the unique solution to a singular elliptic problem with nonlinear convection term and singular weight, Advanced Nonlinear Studies, Volume 8 (2008) no. 2, pp. 391-400 | DOI:10.1515/ans-2008-0209 | Zbl:1168.35370
Chunlian Liu; Zuodong Yang Existence of large solutions for a quasilinear elliptic problem via explosive sub-supersolutions, Applied Mathematics and Computation, Volume 199 (2008) no. 2, pp. 414-424 | DOI:10.1016/j.amc.2007.10.009 | Zbl:1141.35025
Florica Corina Cîrstea; Cristina Trombetti On the Monge-Ampère equation with boundary blow-up: existence, uniqueness and asymptotics, Calculus of Variations and Partial Differential Equations, Volume 31 (2008) no. 2, pp. 167-186 | DOI:10.1007/s00526-007-0108-7 | Zbl:1148.35022
Santiago Cano-Casanova; Julián López-Gómez Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, Journal of Differential Equations, Volume 244 (2008) no. 12, pp. 3180-3203 | DOI:10.1016/j.jde.2007.11.012 | Zbl:1149.34020
Jorge García-Melián Large solutions for an elliptic system of quasilinear equations, Journal of Differential Equations, Volume 245 (2008) no. 12, pp. 3735-3752 | DOI:10.1016/j.jde.2008.04.004 | Zbl:1169.35021
Zhijun Zhang; Ling Mi; Xiugui Yin Blow-up rate of the unique solution for a class of one-dimensional problems on the half-line, Journal of Mathematical Analysis and Applications, Volume 348 (2008) no. 2, pp. 797-805 | DOI:10.1016/j.jmaa.2008.07.074 | Zbl:1176.34033
Zhijun Zhang A boundary blow-up elliptic problem with an inhomogeneous term, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 68 (2008) no. 11, pp. 3428-3438 | DOI:10.1016/j.na.2007.03.034 | Zbl:1158.35040
Tiancheng Ouyang; Zhifu Xie The exact boundary blow-up rate of large solutions for semilinear elliptic problems, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 68 (2008) no. 9, pp. 2791-2800 | DOI:10.1016/j.na.2007.02.026 | Zbl:1138.35025
Chunlian Liu; Zuodong Yang Boundary blow-up quasilinear elliptic problems of the Bieberbach type with nonlinear gradient terms, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 69 (2008) no. 12, pp. 4380-4391 | DOI:10.1016/j.na.2007.10.060 | Zbl:1159.35034
Zhijun Zhang Boundary behavior of solutions to some singular elliptic boundary value problems, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 69 (2008) no. 7, pp. 2293-2302 | DOI:10.1016/j.na.2007.08.009 | Zbl:1151.35032
Florica Corina Cîrstea; Yihong Du Large solutions of elliptic equations with a weakly superlinear nonlinearity, Journal d'Analyse Mathématique, Volume 103 (2007), pp. 261-277 | DOI:10.1007/s11854-008-0008-6 | Zbl:1186.35059
Florica Corina Cîrstea; Yihong Du Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity, Journal of Functional Analysis, Volume 250 (2007) no. 2, pp. 317-346 | DOI:10.1016/j.jfa.2007.05.005 | Zbl:1220.35046
Ahmed Mohammed Boundary asymptotic and uniqueness of solutions to the $p$ -Laplacian with infinite boundary values, Journal of Mathematical Analysis and Applications, Volume 325 (2007) no. 1, pp. 480-489 | DOI:10.1016/j.jmaa.2006.02.008 | Zbl:1142.35412
Jorge García-Melián A remark on uniqueness of large solutions for elliptic systems of competitive type, Journal of Mathematical Analysis and Applications, Volume 331 (2007) no. 1, pp. 608-616 | DOI:10.1016/j.jmaa.2006.09.006 | Zbl:1131.35015
Zongming Guo; Junli Shang Remarks on uniqueness of boundary blow-up solutions, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 66 (2007) no. 2, pp. 484-497 | DOI:10.1016/j.na.2005.11.042 | Zbl:1159.35330
Zhijun Zhang Boundary blow-up elliptic problems of Bieberbach and Rademacher type with nonlinear gradient terms, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 67 (2007) no. 3, pp. 727-734 | DOI:10.1016/j.na.2006.06.025 | Zbl:1131.35026
Jorge García-Melián Uniqueness for boundary blow-up problems with continuous weights, Proceedings of the American Mathematical Society, Volume 135 (2007) no. 9, pp. 2785-2793 | DOI:10.1090/s0002-9939-07-08822-3 | Zbl:1146.35036
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
Florica-Corina Cîrstea; Vicenţiu Rădulescu Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type, Transactions of the American Mathematical Society, Volume 359 (2007) no. 7, pp. 3275-3286 | DOI:10.1090/s0002-9947-07-04107-4 | Zbl:1134.35039
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
Zongming Guo; Feng Zhou Exact multiplicity for boundary blow-up solutions, Journal of Differential Equations, Volume 228 (2006) no. 2, pp. 486-506 | DOI:10.1016/j.jde.2006.02.012 | Zbl:1139.35345
Zhijun Zhang Boundary blow-up elliptic problems with nonlinear gradient terms, Journal of Differential Equations, Volume 228 (2006) no. 2, pp. 661-684 | DOI:10.1016/j.jde.2006.02.003 | Zbl:1130.35063
Zongming Guo; J. R. L. Webb Structure of boundary blow-up solutions for quasi-linear elliptic problems. II: Small and intermediate solutions, Journal of Differential Equations, Volume 211 (2005) no. 1, pp. 187-217 | DOI:10.1016/j.jde.2004.06.008 | Zbl:1134.35339
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
Florica-Corina Cîrstea An extreme variation phenomenon for some nonlinear elliptic problems with boundary blow-up, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 339 (2004) no. 10, pp. 689-694 | DOI:10.1016/j.crma.2004.10.005 | Zbl:1133.35352
Florica-Corina Cîrstea; Vicenţiu Rădulescu Extremal singular solutions for degenerate logistic-type equations in anisotropic media, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 339 (2004) no. 2, pp. 119-124 | DOI:10.1016/j.crma.2004.04.025 | Zbl:1101.35034
Manuel Delgado; Julián López-Gómez; Antonio Suárez Singular boundary value problems of a porous media logistic equation, Hiroshima Mathematical Journal, Volume 34 (2004) no. 1, pp. 57-80 | DOI:10.32917/hmj/1150998071 | Zbl:1112.35065
Yihong Du; Zongming Guo Uniqueness and layer analysis for boundary blow up solutions, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 83 (2004) no. 6, pp. 739-763 | DOI:10.1016/j.matpur.2004.01.006 | Zbl:1081.35032
Florica-Corina Cîrstea; Vicenţiu Rădulescu Asymptotics for the blow-up boundary solution of the logistic equation with absorption., Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 336 (2003) no. 3, pp. 231-236 | DOI:10.1016/s1631-073x(03)00027-x | Zbl:1068.35035
Marius Ghergu; Vicenţiu D. Rădulescu Bifurcation and asymptotics for the Lane–Emden–Fowler equation., Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 337 (2003) no. 4, pp. 259-264 | DOI:10.1016/s1631-073x(03)00335-2 | Zbl:1073.35087
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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^☆ 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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