Systems of reaction-diffusion equations with spatially distributed hysteresis

TL;DR

This paper investigates reaction-diffusion systems with spatially distributed hysteresis, providing conditions for well-posedness despite discontinuities, and models interactions leading to pattern formation in biological systems.

Contribution

It introduces a framework for analyzing reaction-diffusion equations with spatially distributed hysteresis, ensuring well-posedness through geometric conditions.

Findings

Established sufficient conditions for well-posedness

Characterized hysteresis thresholds geometrically

Applied to biological pattern formation models

Abstract

We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis in the right-hand side. The input of hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the discontinuity of hysteresis. These conditions are formulated in terms of geometry of manifolds defining hysteresis thresholds and the graph of initial data.