Nonlinear metastability for a parabolic system of reaction-diffusion equations

TL;DR

This paper investigates the slow, metastable approach to steady states in a reaction-diffusion system, focusing on the asymptotic behavior as viscosity tends to zero, and derives an ODE governing interface dynamics.

Contribution

It provides a rigorous analysis of metastable dynamics in reaction-diffusion systems and derives an ODE for internal interface motion as viscosity vanishes.

Findings

Metastable solutions approach steady states exponentially slowly.

Derived an ODE describing the motion of internal interfaces.

Analyzed solution dynamics near approximate steady states.

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 its 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 solutions in a neighborhood of a one-parameter family of approximate steady states, and we derive an ODE for the position of the internal interfaces.