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Spatially discrete reaction–diffusion equations with discontinuous hysteresis. / Gurevich, Pavel; Tikhomirov, Sergey.

In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, Vol. 35, No. 4, 01.07.2018, p. 1041-1077.

Research output: Contribution to journal › Article › peer-review

Harvard

Gurevich, P & Tikhomirov, S 2018, 'Spatially discrete reaction–diffusion equations with discontinuous hysteresis', Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, vol. 35, no. 4, pp. 1041-1077. https://doi.org/10.1016/j.anihpc.2017.09.006

APA

Gurevich, P., & Tikhomirov, S. (2018). Spatially discrete reaction–diffusion equations with discontinuous hysteresis. Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 35(4), 1041-1077. https://doi.org/10.1016/j.anihpc.2017.09.006

Vancouver

Gurevich P, Tikhomirov S. Spatially discrete reaction–diffusion equations with discontinuous hysteresis. Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire. 2018 Jul 1;35(4):1041-1077. https://doi.org/10.1016/j.anihpc.2017.09.006

Author

Gurevich, Pavel ; Tikhomirov, Sergey. / Spatially discrete reaction–diffusion equations with discontinuous hysteresis. In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire. 2018 ; Vol. 35, No. 4. pp. 1041-1077.

BibTeX

@article{cf7308a5956f44f686623434dff16d95,
title = "Spatially discrete reaction–diffusion equations with discontinuous hysteresis",
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 at−1/2 as t→∞ and explicitly find the rate a.",
keywords = "Hysteresis, Lattice dynamics, Pattern formation, Rattling, Reaction–diffusion equations, Spatial discretisation",
author = "Pavel Gurevich and Sergey Tikhomirov",
year = "2018",
month = jul,
day = "1",
doi = "10.1016/j.anihpc.2017.09.006",
language = "English",
volume = "35",
pages = "1041--1077",
journal = "Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire",
issn = "0294-1449",
publisher = "Elsevier",
number = "4",

}

RIS

TY - JOUR

T1 - Spatially discrete reaction–diffusion equations with discontinuous hysteresis

AU - Gurevich, Pavel

AU - Tikhomirov, Sergey

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

KW - Hysteresis

KW - Lattice dynamics

KW - Pattern formation

KW - Rattling

KW - Reaction–diffusion equations

KW - Spatial discretisation

UR - http://www.scopus.com/inward/record.url?scp=85032905728&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2017.09.006

DO - 10.1016/j.anihpc.2017.09.006

M3 - Article

AN - SCOPUS:85032905728

VL - 35

SP - 1041

EP - 1077

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

SN - 0294-1449

IS - 4

ER -

ID: 43392860