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.
CITATION STYLE
Curran, M., Gurevich, P., & Tikhomirov, S. (2016). Recent advances in reaction-diffusion equations with non-ideal relays. In Understanding Complex Systems (pp. 211–234). Springer Verlag. https://doi.org/10.1007/978-3-319-28028-8_11
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