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

TL;DR

This paper investigates the conditions under which solutions to the Keller-Rubinow model for Liesegang rings are unique in the fast reaction limit, identifying scenarios of non-uniqueness and conditions for solution continuation.

Contribution

It provides an explicit description of the precipitation boundary and characterizes scenarios leading to non-uniqueness, introducing conditions for solution uniqueness over time.

Findings

Solutions are unique for a finite time interval.

Two non-uniqueness scenarios: spontaneous precipitation and entanglement.

Uniqueness depends on transversality conditions in space or time.

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.