Well-posedness of parabolic equations containing hysteresis with diffusive thresholds |
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Authors: | Pavel Gurevich Dmitrii Rachinskii |
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Affiliation: | 1. Freie Universit?t Berlin, Kaiserswerther Str. 16-18, 14195, Berlin, Germany 2. Peoples’ Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198, Russia 3. Department of Mathematical Sciences, University of Texas at Dallas, FO 35, 800 West Campbell Road, Richardson, TX, 75080-3021, USA 4. Department of Applied Mathematics, University College Cork, Western Road, Cork, Ireland
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Abstract: | 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. |
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