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 $$ \partial_t u-Δu=a u-b(x) u^p \text{in} Ω\times \R^+, u(0)=u_0, u(t)|_{\partial Ω}=0 $$ as $p\to +\infty$, where $Ω$ 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