Slow dynamics in reaction-diffusion systems

TL;DR

This paper studies slow, metastable dynamics in reaction-diffusion systems, analyzing layered solutions and deriving an ODE for interface positions as viscosity tends to zero.

Contribution

It provides a rigorous analysis of metastable behavior and derives an ODE governing interface dynamics in reaction-diffusion systems.

Findings

Metastable solutions approach steady states exponentially slowly.

Derived an ODE describing the motion of internal interfaces.

Analyzed layered solutions far from stable configurations.

Abstract

We consider a system of reaction-diffusion equations in a bounded interval of the real line, with emphasis on the metastable dynamics, whereby the time-dependent solution approaches the steady state in an asymptotically exponentially long time interval as the viscosity coefficient $\varepsilon>0$ goes to zero. To rigorous describe such behavior, we analyze the dynamics of layered solutions localized far from the stable configurations of the system, and we derive an ODE for the position of the internal interfaces.