Reaction-diffusion on a time-dependent interval: refining the notion of 'critical length'

TL;DR

This paper investigates reaction-diffusion equations on a time-dependent interval, revealing that solutions' long-term behavior depends on conditions beyond just the domain length, challenging the classical notion of a fixed critical length.

Contribution

It extends the understanding of reaction-diffusion dynamics to time-varying domains, providing new criteria for solution convergence or divergence.

Findings

Solutions can tend to zero or not, even when domain length is below the critical length.

Derived upper and lower estimates for solutions on general time-dependent intervals.

Demonstrated that domain length alone does not determine long-term behavior.

Abstract

A reaction-diffusion equation is studied in a time-dependent interval whose length varies with time. The reaction term is either linear or of KPP type. On a fixed interval, it is well-known that if the length is less than a certain critical value then the solution tends to zero. When the domain length may vary with time, we prove conditions under which the solution does and does not converge to zero in long time. We show that, even with the length always strictly less than the 'critical length', either outcome may occur. Examples are given. The proof is based on upper and lower estimates for the solution, which are derived in this paper for a general time-dependent interval.