Systems of reaction-diffusion equations with spatially distributed hysteresis

Pavel Gurevich and Sergey Tikhomirov

Contact the author: Please use for correspondence this email.
Submission date: 27. Sep. 2013
Pages: 16
published in: Mathematica bohemica, 139 (2014) 2, p. 239-257 
Bibtex
MSC-Numbers: 35K57, 35K45, 47J40
Keywords and phrases: spatially distributed hysteresis, reaction-diffusion equations, wellposedness
Download full preprint: PDF (238 kB)

Abstract:
We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis in the right-hand side. The input of 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 discontinuity of hysteresis. These conditions are formulated in terms of geometry of manifolds defining hysteresis thresholds and the graph of initial data.