Spatially discrete reaction–diffusion equations with discontinuous hysteresis

  • Pavel Gurevich

    Free University of Berlin, Germany; RUDN University, Russia
  • Sergey Tikhomirov

    Saint-Petersburg State Univeristy, Russia

Abstract

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 as and explicitly find the rate .

Cite this article

Pavel Gurevich, Sergey Tikhomirov, Spatially discrete reaction–diffusion equations with discontinuous hysteresis. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 4, pp. 1041–1077

DOI 10.1016/J.ANIHPC.2017.09.006