Spatially discrete reaction–diffusion equations with discontinuous hysteresis
Pavel Gurevich, Sergey B. Tikhomirov · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2017
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\rightarrow \infty and explicitly find the rate a .