Spatially discrete reaction-diffusion equations with discontinuous hysteresis
Pavel Gurevich and Sergey Tikhomirov
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Submission date: 15. Apr. 2015
published in: Annales de l'Institut Henri Poincaré / C (2017), pp not yet known
DOI number (of the published article): 10.1016/j.anihpc.2017.09.006
MSC-Numbers: 37L60, 34C55, 35B36
Keywords and phrases: spatially distributed hysteresis, lattice dynamical systems, pattern formation, rattling
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We deal with a class of lattice dynamical systems, namely hysteretic reaction-diffusion equations that are continuous in time and discrete in space. Our primary goal is to analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling. The rattling is characterized by a specific behavior of the solution profile: while evolving in time, it oscillates up- and downwards and "switches" hysteresis at nodes satisfying a certain rule. In the one-dimensional case, this switching rule makes the profile shape two hills propagating outwards the origin with decreasing velocity. In a prototype case, we prove the existence of rattling and identify the propagation velocity.