Slow motion for gradient systems with equal depth multiple-well potentials

TL;DR

This paper extends the understanding of slow-moving fronts in scalar reaction-diffusion systems with multiple wells to systems, using localized energy evolution to relax initial data assumptions.

Contribution

It introduces a new method based on localized energy analysis to study front speeds in reaction-diffusion systems with multiple wells, relaxing previous initial condition constraints.

Findings

Front speeds are exponentially small in systems with multiple wells.

The method relaxes initial data assumptions compared to previous scalar cases.

Provides a framework for analyzing front dynamics in reaction-diffusion systems.

Abstract

For scalar reaction-diffusion in one space dimension, it is known for a long time that fronts move with an exponentially small speed for potentials with several distinct mini- mizers. The purpose of this paper is to provide a similar result in the case of systems. Our method relies on a careful study of the evolution of localized energy. This approach has the advantage to relax the preparedness assumptions on the initial datum.