MATHEMATICA BOHEMICA, Vol. 139, No. 2, pp. 239-257, 2014

Systems of reaction-diffusion equations with spatially distributed hysteresis

Pavel Gurevich, Sergey Tikhomirov

Pavel Gurevich, Free University Berlin, Arnimallee 3, Berlin, 14195, Germany; Peoples' Friendship University, Mikluho-Maklaya str. 6, Moscow, 117198, Russia, e-mail:; Sergey Tikhomirov, Chebyshev Laboratory, Saint-Petersburg State University, 14th line of Vasilievsky island, 29B, Saint-Petersburg, 199178, Russia; Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany, e-mail:

Abstract: We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the assumed discontinuity of hysteresis. These conditions are formulated in terms of geometry of the manifolds defining the hysteresis thresholds and the spatial profile of the initial data.

Keywords: spatially distributed hysteresis; reaction-diffusion equation; well-posedness

Classification (MSC 2010): 35K57, 35K45, 47J40

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