Reaction-diffusion equation with spatially distributed hysteresis

We consider continuous and discrete reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. In particular, such problems describe chemical reactions and biological processes in which diffusive and nondiffusive substances interact according to hysteresis law. As a result various spatial and spatio-temporal patterns may appear.

We found a broad class of initial data (transverse functions) for which a solution exists, is unique, and continuously depends on initial data. As far as we know, this is the first uniqueness result for such systems with hysteresis.

For the nontransverse case we consider the discretization of the problem and consider it as a systems of coupled differential equation with hysteresis. We analyse patterns which appears in this system and leads to nonexistence of a solution for the original reaction-diffusion equation. It is interesting to mention that for 2-dimensional space those patterns depends on the type of discretization.

This is joint work with P. Gurevich.