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Spatially distributed hysteresis in reaction-diffusion equations

Dr. Pavel Gurevich:


We consider reaction-diffusion equations involving a hysteretic discontinuity which is defined at each spatial point. These problems describe chemical reactions and biological processes in which diffusive and nondiffusive substances interact according to hysteresis law. As an example, one can think of nondiffusive bacteria and diffusive nutrient or nondiffusive cells and diffusive proteins, or nondiffusive neurons and diffusive extracellular ions, etc.

In the talk, we discuss a relevant mathematical setting. We show that the spatially distributed hysteresis may switch at different spatial points at different time moments, dividing the spatial domain into subdomains where hysteresis has the same state and thus defining spatial topology of hysteresis. The boundaries between the subdomains are free boundaries whose motion depends both on the reaction-diffusion equation and hysteresis. The interplay of those two leads to formation of different spatio-temporal patterns.

As the first step towards the rigorous theory, we will show that the problem has a unique solution as long as this solution preserves the spatial topology of hysteresis, while the change of topology may occur only via a spatial nontransversality of the solution. In the end, we will formulate open questions for further research.

The results are obtained jointly with R. Shamin and S. Tikhomirov.


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