Geodesic
Well-posedness of parabolic equations containing hysteresis with diffusive thresholds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 92-114

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We study complex systems arising, in particular, in population dynamics, developmental biology, and bacterial metabolic processes, in which each individual element obeys a relatively simple hysteresis law (a non-ideal relay). Assuming that hysteresis thresholds fluctuate, we consider the arising reaction-diffusion system. In this case, the spatial variable corresponds to the hysteresis threshold. We describe the collective behavior of such a system in terms of the Preisach operator with time-dependent measure which is a part of the solution for the whole system. We prove the well-posedness of the system and discuss the long-term behavior of solutions. 
@article{TM_2013_283_a6,
     author = {Pavel Gurevich and Dmitrii Rachinskii},
     title = {Well-posedness of parabolic equations containing hysteresis with diffusive thresholds},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {92--114},
     publisher = {mathdoc},
     volume = {283},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_283_a6/}
}
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Pavel Gurevich; Dmitrii Rachinskii. Well-posedness of parabolic equations containing hysteresis with diffusive thresholds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 92-114. http://geodesic.mathdoc.fr/item/TM_2013_283_a6/