TY - JOUR

T1 - Spatially discrete reaction–diffusion equations with discontinuous hysteresis

AU - Gurevich, Pavel

AU - Tikhomirov, Sergey

PY - 2018/7/1

Y1 - 2018/7/1

N2 - We address the question: Why may reaction–diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order at−1/2 as t→∞ and explicitly find the rate a.

AB - We address the question: Why may reaction–diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order at−1/2 as t→∞ and explicitly find the rate a.

KW - Hysteresis

KW - Lattice dynamics

KW - Pattern formation

KW - Rattling

KW - Reaction–diffusion equations

KW - Spatial discretisation

UR - http://www.scopus.com/inward/record.url?scp=85032905728&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2017.09.006

DO - 10.1016/j.anihpc.2017.09.006

M3 - Article

AN - SCOPUS:85032905728

VL - 35

SP - 1041

EP - 1077

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

SN - 0294-1449

IS - 4

ER -