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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.
@article{AIHPC_2018__35_4_1041_0,
author = {Gurevich, Pavel and Tikhomirov, Sergey},
title = {Spatially discrete reaction{\textendash}diffusion equations with discontinuous hysteresis},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1041--1077},
publisher = {Elsevier},
volume = {35},
number = {4},
year = {2018},
doi = {10.1016/j.anihpc.2017.09.006},
mrnumber = {3795026},
zbl = {1391.34026},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.09.006/}
}
TY - JOUR AU - Gurevich, Pavel AU - Tikhomirov, Sergey TI - Spatially discrete reaction–diffusion equations with discontinuous hysteresis JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 1041 EP - 1077 VL - 35 IS - 4 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.09.006/ DO - 10.1016/j.anihpc.2017.09.006 LA - en ID - AIHPC_2018__35_4_1041_0 ER -
%0 Journal Article %A Gurevich, Pavel %A Tikhomirov, Sergey %T Spatially discrete reaction–diffusion equations with discontinuous hysteresis %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 1041-1077 %V 35 %N 4 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.09.006/ %R 10.1016/j.anihpc.2017.09.006 %G en %F AIHPC_2018__35_4_1041_0
Gurevich, Pavel; Tikhomirov, Sergey. Spatially discrete reaction–diffusion equations with discontinuous hysteresis. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 1041-1077. doi: 10.1016/j.anihpc.2017.09.006
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