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.

Original languageEnglish
Pages (from-to)1041-1077
Number of pages37
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume35
Issue number4
DOIs
StatePublished - 1 Jul 2018

    Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

    Research areas

  • Hysteresis, Lattice dynamics, Pattern formation, Rattling, Reaction–diffusion equations, Spatial discretisation

ID: 43392860