Breakdown of Liesegang precipitation bands in a simplified fast reaction limit of the Keller-Rubinow model

Abstract

We study solutions to the integral equation \[ \omega(x) = \Gamma - x^2 \int_{0}^1 K(\theta) \, H(\omega(x\theta)) \, \mathrm d \theta \] where $\Gamma>0$, $K$ is a weakly degenerate kernel satisfying, among other properties, $K(\theta) \sim k \, (1-\theta)^\sigma$ as $\theta \to 1$ for constants $k>0$ and $\sigma \in (0, \log_2 3 -1)$, $H$ denotes the Heaviside function, and $x \in [0,\infty)$. This equation arises from a reaction-diffusion equation describing Liesegang precipitation band patterns under certain simplifying assumptions. We argue that the integral equation is an analytically tractable paradigm for the clustering of precipitation rings observed in the full model. This problem is nontrivial as the right hand side fails a Lipschitz condition so that classical contraction mapping arguments do not apply. Our results are the following. Solutions to the integral equation, which initially feature a sequence of relatively open intervals on which $\omega$ is positive ("rings") or negative ("gaps") break down beyond a finite interval $[0,x^*]$ in one of two possible ways. Either the sequence of rings accumulates at $x^*$ ("non-degenerate breakdown") or the solution cannot be continued past one of its zeroes at all ("degenerate breakdown"). Moreover, we show that degenerate breakdown is possible within the class of kernels considered. Finally, we prove existence of generalized solutions which extend the integral equation past the point of breakdown.