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Z. Darbenas, R. van der Hout, and M. Oliver,
Conditional uniqueness of solutions to the Keller-Rubinow model for Liesegang rings in the fast reaction limit,
submitted for publication.
Abstract:
We study the question of uniqueness of weak solution to the fast
reaction limit of the Keller and Rubinow model for Liesegang rings as
introduced by Hilhorst et al. (J. Stat. Phys. 135, 2009,
pp 107-132). The model is characterized by a discontinuous
reaction term which can be seen as an instance of spatially
distributed non-ideal relay hysteresis. In general, uniqueness of
solutions for such models is conditional on certain transversality
conditions. For the model studied here, we give an explicit
description of the precipitation boundary which gives rise to two
scenarios for non-uniqueness, which we term "spontaneous
precipitation" and "entanglement". Spontaneous precipitation can
be easily dismissed by an additional, physically reasonable criterion
in the concept of weak solution. The second scenario is one where the
precipitation boundaries of two distinct solutions cannot be ordered
in any neighborhood of some point on their common precipitation
boundary. We show that for a finite, possibly short interval of time,
solutions are unique. Beyond this point, unique continuation is
subject to a spatial or temporal transversality condition. The
temporal transversality condition takes the same form that would be
expected for a simple multicomponent semilinear ODE with discontinuous
reaction terms.
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