Self-Similar Pattern in Coupled Parabolic Systems as Non-Equilibrium Steady States

TL;DR

This paper investigates reaction-diffusion and dissipative systems on unbounded domains, revealing that they exhibit self-similar steady states due to persistent mass or energy flow, despite local equilibrium formation.

Contribution

It introduces the concept of self-similar patterns in coupled parabolic systems as non-equilibrium steady states, extending understanding of their long-term behavior.

Findings

Systems reach local equilibrium quickly but maintain persistent flow.

Rescaled variables reveal convergence to self-similar steady states.

The approach links finite domain Lyapunov functions to infinite domain behavior.

Abstract

We consider reaction-diffusion systems and other related dissipative systems on unbounded domains which would have a Liapunov function (and gradient structure) when posed on a finite domain. In this situation, the system may reach local equilibrium on a rather fast time scale but the infinite amount of mass or energy leads to persistent mass or energy flow for all times. In suitably rescaled variables the system converges to a steady state that corresponds to asymptotically self-similar behavior in the original system.