Spatially discrete reaction–diffusion equations with discontinuous hysteresis

Gurevich, Pavel; Tikhomirov, Sergey

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 1041-1077.

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.

Mots clés : Hysteresis, Pattern formation, Reaction–diffusion equations, Rattling, Spatial discretisation, Lattice dynamics

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