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## The Hysteretic Limit of a Reaction-Diffusion System with a Small Parameter

Abstract:

In this talk I will consider a reaction-diffusing system comprising one

diffusing substance and a non-diffusing ensemble of hysteresis

operators. Such equations model a variety of biological processes where

the diffusion rate is slow compared to the reaction speed of the

non-diffusing species. The hysteresis operators will either be solutions

to an ODE with a small parameter, or their formal limit obtained by

setting the parameter equal to zero. The later is a discontinuous

operator and it is well known that for a single operator it indeed

describes the limiting behavior of the ODE as the parameter approaches

zero. The corresponding behavior in the PDE setting has only been

observed numerically. In this talk, I will present the first rigorous

convergence results in the PDE setting, in particular, I will show

asymptotics with respect to the small parameter and describe how they

relate to the spatial regularity of the diffusing substance.