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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Gurevich, P., Rachinskii, D. Well-posedness of parabolic equations containing hysteresis with diffusive thresholds. Proc. Steklov Inst. Math. 283, 87–109 (2013). https://doi.org/10.1134/S0081543813080075
- Parabolic Equation
- STEKLOV Institute
- Neumann Boundary Condition
- Switching Threshold