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Increasing Powers in a Degenerate Parabolic Logistic Equation

  • José Francisco Rodrigues
  • Hugo Tavares

Abstract

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem
$$\partial_t u-\Delta u=a u-b(x) u^p \quad\text{in } \varOmega\times \mathbb{R} ^+,\quad u(0)=u_0,\quad u(t)|_{\partial \varOmega}=0, $$
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.

Keywords

Parabolic logistic equation Obstacle problem Positive solution Increasing power, subsolution and supersolution 

Mathematics Subject Classification

35B40 35B09 35K91 

Notes

Acknowledgements

The second author would like to thank Jesús Hernández and Pedro Freitas for the useful discussions related with this paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and CMAFUniversidade de LisboaLisboaPortugal

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