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Systems of reaction-diffusion equations with spatially distributed hysteresis. (English) Zbl 1340.35136
Summary: We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the 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 assumed discontinuity of hysteresis. These conditions are formulated in terms of geometry of the manifolds defining the hysteresis thresholds and the spatial profile of the initial data.

MSC:
35K57 Reaction-diffusion equations
35K45 Initial value problems for second-order parabolic systems
47J40 Equations with nonlinear hysteresis operators
35B36 Pattern formations in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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