Well-posedness of parabolic equations containing hysteresis with diffusive thresholds

TL;DR

This paper investigates reaction-diffusion systems with hysteresis thresholds, proving well-posedness and analyzing long-term behavior, with applications in biology and population dynamics.

Contribution

It introduces a novel approach to modeling systems with fluctuating hysteresis thresholds using the Preisach operator, establishing well-posedness and long-term dynamics.

Findings

Proved well-posedness of the hysteresis-based reaction-diffusion system

Described the collective behavior via the Preisach operator with time-dependent measure

Discussed the long-term behavior of solutions

Abstract

We study complex systems arising, in particular, in population dynamics, developmental biology, and bacterial metabolic processes, in which each individual element obeys a relatively simple hysteresis law (a non-ideal relay). Assuming that hysteresis thresholds fluctuate, we consider the arising reaction-diffusion system. In this case, the spatial variable corresponds to the hysteresis threshold. We describe the collective behavior of such a system in terms of the Preisach operator with time-dependent measure which is a part of the solution for the whole system. We prove the well-posedness of the system and discuss the long-term behavior of solutions.