Well-posedness of parabolic equations containing hysteresis with diffusive thresholds

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• 作者：Pavel Gurevich (1) (2)
  Dmitrii Rachinskii (3) (4)
  
• 刊名：Proceedings of the Steklov Institute of Mathematics
• 出版年：2013
• 出版时间：December 2013
• 年：2013
• 卷：283
• 期：1
• 页码：87-109
• 全文大小：1,338 KB
• 作者单位：Pavel Gurevich (1) (2)
  Dmitrii Rachinskii (3) (4)
  
  1. Freie Universit盲t Berlin, Kaiserswerther Str. 16-18, 14195, Berlin, Germany
  2. Peoples鈥?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
  
• ISSN：1531-8605

文摘

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.