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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
35/2015

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

Marc Curran, Pavel Gurevich and Sergey Tikhomirov

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.

Received:
Jul 11, 2015
Published:
Jul 15, 2015
MSC Codes:
35K57, 35B36, 35K20
Keywords:
spatially distributed hysteresis, reaction-diffusion equation, well-posedness, spatial discretisation, rattling

Related publications

inBook
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)