Well-posedness of parabolic equations containing hysteresis with diffusive thresholds

  • Pavel GurevichEmail author
  • Dmitrii Rachinskii


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.


Lactose Parabolic Equation STEKLOV Institute Neumann Boundary Condition Switching Threshold 
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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany
  2. 2.Peoples’ Friendship University of RussiaMoscowRussia
  3. 3.Department of Mathematical SciencesUniversity of Texas at DallasRichardsonUSA
  4. 4.Department of Applied MathematicsUniversity College CorkCorkIreland

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