Generation of interface for an Allen-Cahn equation with nonlinear diffusion

TL;DR

This paper studies a nonlinear diffusion equation with a bistable reaction term, analyzing how interfaces form rapidly when the reaction coefficient becomes very large, with implications for population dynamics models.

Contribution

It proves the generation of interface property for a broad class of initial data as the reaction coefficient approaches infinity.

Findings

Interface generation occurs rapidly for large reaction coefficients.

The analysis applies to general initial data in population dynamics models.

The results provide insights into the behavior of solutions in the limit of strong reactions.

Abstract

In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property.