Recent Advances in Reaction-Diffusion Equations with Non-Ideal Relays

Marc Curran, Pavel Gurevich, and Sergey Tikhomirov

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Submission date: 11. Jul. 2015
Pages: 26
published in: Control of self-organizing nonlinear systems / E. Schöll ... (eds.)
[Cham] : Springer International Publishing, 2016. - P. 211 - 234
(Understanding complex systems) 
DOI number (of the published article): 10.1007/978-3-319-28028-8_11
Bibtex
MSC-Numbers: 35K57, 35B36, 35K20
Keywords and phrases: spatially distributed hysteresis, reaction-diffusion equation, well-posedness, spatial discretisation, rattling
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Abstract:
We survey recent results on reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time, and also describe its regularity. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism — rattling, which indicates how one should reset the continuous model to make it well posed.