Chinese Annals of Mathematics, Series B

, Volume 34, Issue 2, pp 277–294 | Cite as

Increasing powers in a degenerate parabolic logistic equation

  • José Francisco RodriguesEmail author
  • Hugo Tavares


The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem
$$\partial _t u - \Delta u = au - b\left( x \right)u^p in \Omega \times \mathbb{R}^ + , u(0) = u_0 , \left. {u(t)} \right|_{\partial \Omega } = 0,$$
as p → + ∞, where Ω is a bounded domain, and b(x) is a nonnegative function. The authors deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards fully describe its long time behavior.


Parabolic logistic equation Obstacle problem Positive solution Increasing power Subsolution and supersolution 

2000 MR Subject Classification

35B40 35B09 35K91 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Lions, J. L., Quelques Méthodes de Résolution des Problémes aux Limites Nonlinéaires, Gauthier-Villars, Paris, 1969.Google Scholar
  2. [2]
    Zheng, S., Nonlinear evolutions equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133, Chapmam & Hall/CRC, Boca Raton, FL, 2004.CrossRefGoogle Scholar
  3. [3]
    Dancer, E. and Du, Y., On a free boundary problem arising from population biology, Indiana Univ. Math. J., 52, 2003, 51–67.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Dancer, E., Du, Y. and Ma, L., Asymptotic behavior of positive solutions of some elliptic problems, Pacific Journal of Mathematics, 210, 2003, 215–228.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Dancer, E., Du, Y. and Ma, L., A uniqueness theorem for a free boundary problem, Proc. Amer. Math. Soc., 134, 2006, 3223–3230.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Fraile, J. M., Koch Medina, P., López-Gómez J., et al., Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127, 1996, 295–319.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Du, Y. and Guo, Z., The degenerate logistic model and a singularly mixed boundary blow-up problem, Discrete and Continuous Dyn. Syst., 14, 2006, 1–29.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Boccardo, L. and Murat, F., Increase of power leads to bilateral problems, Composite Media and Homogenization Theory, G. Dal Maso and G. F. Dell’Antonio (eds.), World Scientific, Singapore, 1995, 113–123.Google Scholar
  9. [9]
    Dall’Aglio, A. and Orsina, L., On the limit of some nonlinear elliptic equations involving increasing powers, Asymptotic Analysis, 14, 1997, 49–71.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Ladyzenskaja, O., Solonnikov, V. and Uralceva, N., Linear and Quasi-linear Equations of Parabolic Type, A. M. S., Providence, RI, 1988.Google Scholar
  11. [11]
    Guo, Z. and Ma, L., Asymptotic behavior of positive solutions of some quasilinear elliptic problems, J. London Math. Soc., 76, 2007, 419–437.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Grossi, M., Asymptotic behaviour of the Kazdan-Warner solution in the annulus, J. Differential Equations, 223, 2006, 96–111.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Grossi, M. and Noris, B., Positive constrained minimizers for supercritical problems in the ball, Proc. Amer. Math. Soc., 140, 2012, 2141–2154.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Bonheure, D. and Serra, E., Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth, NoDEA, 18, 2011, 217–235.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Lieberman, G., Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.zbMATHCrossRefGoogle Scholar
  16. [16]
    Rodrigues, J. F., On a class of parabolic unilateral problems, Nonlinear Anal., 10, 1986, 1357–1366.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics and CMAFUniversidade de LisboaLisboaPortugal

Personalised recommendations