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.

Keywords

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

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