Increasing powers in a degenerate parabolic logistic equation

TL;DR

This paper investigates the asymptotic behavior of positive solutions to a degenerate parabolic logistic equation as the exponent p approaches infinity, revealing a connection to a parabolic obstacle problem and analyzing long-term dynamics.

Contribution

It establishes the limiting behavior of solutions as p tends to infinity and characterizes the resulting obstacle problem and its long-term evolution.

Findings

Limit of solutions as p→∞ solves a parabolic obstacle problem

Full description of the long-time behavior of solutions

Connection between logistic equations and obstacle problems

Abstract

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