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MiS Preprint

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

Marc Curran, Pavel Gurevich and Sergey Tikhomirov


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.

MSC Codes:
35K57, 35B36, 35K20
spatially distributed hysteresis, reaction-diffusion equation, well-posedness, spatial discretisation, rattling

Related publications

2016 Repository Open Access
Mark Curran, Pavel Gurevich and Sergey Tikhomirov

Recent advances in reaction-diffusion equations with non-ideal relays

In: Control of self-organizing nonlinear systems / Eckehard Schöll... (eds.)
[Cham] : Springer International Publishing, 2016. - pp. 211-234
(Understanding complex systems)