Increasing powers in a degenerate parabolic logistic equation

Abstract

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.

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Correspondence to José Francisco Rodrigues.

Additional information

In honor of the scientific heritage of Jacques-Louis Lions

Project supported by Fundação para a Ciência e a Tecnologia (FCT) (No. PEst OE/MAT/UI0209/2011). The second author was also supported by an FCT grant (No. SFRH/BPD/69314/201).

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Rodrigues, J.F., Tavares, H. Increasing powers in a degenerate parabolic logistic equation. Chin. Ann. Math. Ser. B 34, 277–294 (2013). https://doi.org/10.1007/s11401-013-0762-3

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Keywords

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

2000 MR Subject Classification

  • 35B40
  • 35B09
  • 35K91