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We study the question of uniqueness of weak solution to the fast reactionlimit of the Keller and Rubinow model for Liesegang rings as introduced byHilhorst et al. (J. Stat. Phys. 135, 2009, pp. 107-132). The model ischaracterized by a discontinuous reaction term which can be seen as an instanceof spatially distributed non-ideal relay hysteresis. In general, uniqueness ofsolutions for such models is conditional on certain transversality conditions.For the model studied here, we give an explicit description of theprecipitation boundary which gives rise to two scenarios for non-uniqueness,which we term "spontaneous precipitation" and "entanglement". Spontaneousprecipitation can be easily dismissed by an additional, physically reasonablecriterion in the concept of weak solution. The second scenario is one where theprecipitation boundaries of two distinct solutions cannot be ordered in anyneighborhood of some point on their common precipitation boundary. We show thatfor a finite, possibly short interval of time, solutions are unique. Beyondthis point, unique continuation is subject to a spatial or temporaltransversality condition. The temporal transversality condition takes the sameform that would be expected for a simple multicomponent semilinear ODE withdiscontinuous reaction terms.

Conditional uniqueness of solutions to the Keller-Rubinow model for Liesegang rings in the fast reaction limit