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.