Recent Advances in Reaction-Diffusion Equations with Non-Ideal Relays

TL;DR

This paper surveys recent developments in reaction-diffusion equations with hysteresis, connecting them to free boundary problems, and introduces a new pattern formation mechanism called rattling.

Contribution

It introduces the concept of spatial transversality, establishes solution uniqueness and regularity, and proposes a discretization approach to analyze nontransverse solutions and pattern formation.

Findings

Unique solutions for transverse initial data.

Introduction of the rattling pattern formation mechanism.

Discretization method for nontransverse solutions.

Abstract

We survey recent results on reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time, and also describe its regularity. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism --- {\it rattling}, which indicates how one should reset the continuous model to make it well posed.