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 $${\partial }_{t}u-\Delta u=au-b(x)u^p\text{in}\Omega \times {\mathbb{R}}^{+},u(0)=u_0,u(t){|}_{\partial \Omega }=0$$ as $p\to +\infty$, where $\Omega$ is a bounded domain and $b(x)$ is a nonnegative function. We deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards we fully describe its long time behavior.

15 pages