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


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 a.

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