Page Content
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.