Interface dynamics of the porous medium equation with a bistable reaction term

TL;DR

This paper analyzes the interface dynamics of a porous medium equation with a bistable reaction term, demonstrating the formation and propagation of transition layers and their convergence to a sharp interface limit governed by degenerate traveling waves.

Contribution

It provides a rigorous analysis of the asymptotic behavior and interface evolution in a porous medium equation with a bistable reaction, linking diffuse interface dynamics to sharp interface limits.

Findings

Rapid formation of transition layers

Propagation of transition layers as the interface evolves

Convergence to a sharp interface limit governed by traveling waves

Abstract

We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.