Spatially discrete reaction–diffusion equations with discontinuous hysteresis

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. © 2017 Elsevier Masson SAS

Authors
Gurevich P. 1, 2 , Tikhomirov S.3
Publisher
Elsevier Masson SAS
Number of issue
4
Language
English
Pages
1041-1077
Status
Published
Volume
35
Year
2018
Organizations
  • 1 Free University of Berlin, Germany
  • 2 RUDN University, Russian Federation
  • 3 Saint-Petersburg State Univeristy, Russian Federation
Keywords
Hysteresis; Lattice dynamics; Pattern formation; Rattling; Reaction–diffusion equations; Spatial discretisation
Date of creation
19.10.2018
Date of change
19.10.2018
Short link
https://repository.rudn.ru/en/records/article/record/6560/