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

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.


Introduction
Reaction-diffusion equations with discontinuous hysteresis occur in a range of modeling problems [3,17,20,25,26]. We are particularly interested in non-ideal relays-two-valued operators where the output switches from the "off-state" 0 to the "on-state" 1 when the input crosses a threshold β, and switches back to zero only when the input drops below a lower threshold α < β. There are different choices to define the behavior of the relay at the threshold. The relay may be restricted to binary values and jump when the threshold is reached or exceeded. Alternatively, the relay may be completed : when the threshold is reached but not exceeded, the relay may take fractional values which can change monotonically in time; when the input drops below the threshold without having crossed, the attained fractional value get "frozen in". See, e.g., [5] for a detailed description of different relay behaviors.
Rigorous mathematical results are of two types. For reaction-diffusion equations with completed relays, weak limit arguments lead to existence of solution [24,2] but not necessarily their uniqueness and continuous dependence on the data. For reaction-diffusion equations with non-completed non-ideal relays, local wellposedness, including uniqueness and continuous dependence, holds true provided that a certain transversality condition on the data is satisfied. The solution can be continued in time for as long as the transversality condition remains satisfied [12,11,5]. We finally remark that for some types of spatially distributed hysteresis, variational approaches may be available [21].
In this paper, we study an explicit example of a reaction-diffusion equation with relay hysteresis which demonstrates that, in general, global-in-time solutions require the notion of a completed relay. Our example is motivated from the study of the fast reaction limit, introduced by Hilhorst et al. [14,15], of the Keller and Rubinow model for Liesegang precipitation rings [16]. This limit model, which we will refer to as the HHMO-model, is a scalar reaction-diffusion equation driven by a point source which is constant in parabolic similarity variables with a reaction term modeled by a relay with a positive upper threshold and zero lower threshold. As a consequence, at a fixed location in space, the reaction, once switched on, can never switch off. The loci of reaction then form a spatial precipitation pattern.
Simple as it seems, an analysis of the HHMO-model faces the same type of difficulty as the analysis of other reaction-diffusion equations with relay hysteresis; in particular, the questions of global uniqueness and continuous dependence on the data remain open. Our aim here is to provide insight into the essential features of the distributed relay dynamics. We make use of a remarkable feature of the HHMOmodel: it can be formally simplified to an equation, different but sufficiently similar to the actual HHMO-model, that is self-similar in parabolic similarity variables. This new model, which we shall refer to as the simplified HHMO-model, reduces to a single scalar integral equation, i.e., can be considered as a scalar dynamical system with memory. The simplified model is finally simple enough that a fairly complete explicit analysis is possible, which is the main contribution of this paper.
We prove that the binary precipitation pattern in the dynamics of the simplified HHMO-model must break down in finite space-time. Beyond the point of breakdown, it can only be continued as a generalized solution. We think of the behavior prior to breakdown as analogous to the well-posedness result for binary switching relays in the spirit of Gurevich et al. [11] and the behavior past the point of breakdown as generalized solutions in the sense of Visintin [24]. While these analogies are tentative and we make no claim that the simplified HHMO-model reflects the behavior of true Liesegang precipitation patterns, the study of this model offers a paradigm for the breakdown of binary patterns. In particular, it gives insight that breakdown can happen in two distinct ways. We believe that more general models-which may not share the symmetry which makes the explicit results of this paper possible-are capable of exhibiting the behaviors observed here, so that the results of this paper provide a lower bound on the complexity which must be addressed when studying more general situations. We also offer a possible perspective for a reformulation of the problem that may lead to well-posedness past the point of breakdown.
Clearly, at x 0 = 0, ω(x 0 ) = Γ > 0 and there must be a point x 1 at which ω changes sign, i.e., where the concentration falls below the super-saturation threshold. Continuing, we may define a sequence x i of loci where ω changes sign, so that (x i , x i+1 ) corresponds to a "ring" or "band" where precipitation occurs when i is even and to a precipitation gap when i is odd. Given the physical background of the problem, we might think that the x i form an unbounded sequence, indicating that the entire domain is covered by a pattern of rings or gaps, or, if the sequence is finite, that the last ring or gap extends to infinity.
Our first result proves that this is not the case: The sequence x i either has a finite accumulation point x * or there is a finite index i such that ω cannot be extended past x * = x i in the sense of equation (1). We call the former case non-degenerate, the latter degenerate.
Our second result demonstrates the existence of degenerate solutions to (1). To this end, we present the construction of a kernel where the solution cannot be continued past the first gap, i.e., where the point of breakdown is x * = x 2 .
To extend the solution past x * , we introduce the concept of extended solutions, reflecting the concept of a completed relay in the spirit of [24] and also [15]. Extended solutions are pairs (ω, ρ) where ω ∈ C([0, ∞)) and subject to the condition that ρ takes values from the Heaviside graph, i.e., As our third result, we prove existence of extended solutions. Extended solutions are unique under the condition that they are regularly extended, namely that ω remains identically zero on some right neighborhood [x * , b) past the point of breakdown. The remainder of the paper is structured as follows. In Section 2, we recall some background on Liesegang rings and the fast reaction limit of the Keller-Rubinow model. In Section 3, we simplify the model to the scalar integral equation (1) and present arguments and numerical evidence that the simplified model reflects the qualitative behavior of the full model. We then proceed to show, in Section 4, that the sequence of precipitation bands either terminates finitely or has a finite accumulation point. In Section 5, we provide a construction that shows that within the class of kernels considered, finite termination is possible. Section 6 discusses extended solutions in the sense of (2). We conclude with a brief discussion and outlook.
2. The Keller-Rubinow model in the fast reaction limit Liesegang precipitation bands are structured patterns in reaction-diffusion kinetics which emerge when, in a chain of two chemical reactions, the second reaction is triggered upon exceeding a supersaturation threshold and is maintained until the reactant concentration falls below a lower so-called saturation threshold. Within suitable parameter ranges, the second reaction will only ignite in restricted spatial regions. When the product of the final reaction precipitates, these regions may be visible as "Liesegang rings" or "Liesegang bands" in reference to German chemist Raphael Liesegang who described this phenomenon in 1896. For a review of the history and chemistry of Liesegang patterns, see [13,23].
Keller and Rubinow [16] gave a quantitative model of Liesegang bands in terms of coupled reaction-diffusion equations; see, in particular, Duley et al. [10] for a comprehensive recent study of this model by asymptotic and numerical methods. We note that there is a competing description in terms of competitive growth of precipitation germs [22] which will not play any role in the following; see, e.g., [18] for a comparative discussion.
Our starting point is the fast reaction limit of the Keller-Rubinow model, where the first-stage reaction rate constant is taken to infinity and one of the first-stage reactant is assumed to be immobile. Hilhorst et al. [14,15] proved that, in this limit, the first-stage reaction can be solved explicitly and contributes a point source of reactant for the second-stage process. Thus, only one scalar reaction-diffusion equation for the second-stage reactant concentration u = u(x, t) remains. Formulated on the half-line, the fast reaction limit, which we shall refer to as the full HHMO-model, reads as follows: where α and β are positive constants and the precipitation function p[x, t; u] is constrained by In this expression, u * > 0 is the super-saturation threshold, i.e., the ignition threshold for the second-stage reaction. For simplicity, the saturation threshold is taken to be zero. This means that once the reaction is ignited at some spatial location x, it will not ever be extinguished at x. Hilhorst et al. [15] proved existence of weak solutions to (4); the question of uniqueness was left open. It is important to note that a weak solution is always a tuple (u, p) where p is constrained, but not defined uniquely in terms of u, by (4d). The analytic difficulties lie in the fact that the onset of precipitation is a free boundary in the (x, t)-plane. Moreover, the precipitation term is discontinuous, so that most of the standard analytical tools are not applicable; in particular, estimates based on energy stability fail. In [6,7], we are able to prove uniqueness for at least an initial short interval of time and derive a sufficient condition for uniqueness at later times. We conjecture that it is possible to obtain instances of non-uniqueness when the problem is considered with arbitrary smooth initial data or smooth additional forcing. One of questions posed in [15] is the problem of proving that the precipitation function p takes only binary values. Our results here suggest that, on the contrary, binary precipitation patterns can only exist in a finite region of space-time.

The simplified HHMO-model
In the following, we detail the connection between the full HHMO-model (4) and the integral equation (1). The key observation is that, when written in a suitable equivalent form, there are only two terms in the full model which do not possess a parabolic scaling symmetry. We cite a mixture of analytic and numerical evidence that suggest that these terms have a negligible impact on the qualitative behavior of the solution: One of the neglected terms represents linear damping toward equilibrium. It is asymptotically subdominant relative to the precipitation term; moreover, its presence could only enhance relaxation to equilibrium. The other term is observed to be asymptotically negligible as the width of the precipitation rings decreases, hence its contribution vanishes as the point of breakdown is approached. Leaving only terms which scale parabolically self-similarly, one of the variables of integration in the Duhamel formula representation of the simplified model can be integrated out, leaving an expression of the form (1) with a complicated, yet explicit expression for the kernel K which is shown, using a mixture of analysis and numerical verification, to satisfy properties (i)-(iii).
Numerical evidence suggests that solutions to the full HHMO-model converge, very rapidly and robustly, to a steady state Φ(η) with respect to the parabolic similarity variable η = x/ √ t as t → ∞, see Figure 1 which is explained in detail further below. A proof of convergence to a steady state is difficult for much the same reasons that well-posedness is difficult, but [9] were able to prove a slightly weaker result: assuming that the HHMO-solution converges to a steady state Φ(η) at all, this steady state must satisfy the differential equation In this formulation, γ is an unknown constant. To determine γ uniquely, this second order system has an additional internal boundary condition (5d) which expresses that the reactant concentration in the HHMO-model converge to the critical value u * at the source point which, in similarity coordinates, moves along the line η = α. There exists a unique solution (Φ, γ) to (5) with γ > 0 and Φ given by where M is Kummer's confluent hypergeometric function [1], κ is a solution of the algebraic equation and γ = κ(κ − 1), subject to the solvability condition In x-t coordinates, the self-similar solution to (5) takes the form Throughout this paper, we assume that α, β, and u * satisfy the solvability condition (8), so that the self-similar solution φ exists. (For triples α, β, and u * which violate the solvability condition, the corresponding weak solution precipitates only a finite region and the asymptotic state is easy to determine explicitly; for details, see [9].) for the full HHMO-model in the form (10). Each subsequent graph zooms into the boxed area of the previous. The simulation parameters are α = β = 1, u * = 0.2, ∆s = ∆η = 5 · 10 −5 .
We now write w = u − φ to denote the difference between the solution of the full HHMO-model (4) and the self-similar profile (9). Then w solves the equation To pass to the simplified HHMO-model, we make two changes to the this equation: (a) Precipitation is triggered on the condition that u > u * on the line x 2 = α 2 t, and (b) the damping term pw in (10a) is neglected. The simplified model then reads The impact of the simplifications can be seen from the simulations shown in Fig  . ω(x) = w(x, w 2 /α 2 ) for the simplified HHMO-model (11). Each subsequent graph zooms into the boxed area of the previous. The simulation parameters are α = β = 1, u * = 0.1, ∆s = ∆η = 3.33 · 10 −5 .
profile Φ, i.e., w(t) → 0 as t → ∞. As expected, due to the lack of the linear damping term pw, the simplified model takes much longer to equilibrize, but the asymptotics remain unchanged. We note that, due to the scaling used, the code is not uniformly accurate in time, so that at every fixed resolution, the large-time behavior is dominated by numerical error and the norm of w will spuriously increase. Increasing the numerical resolution will move the point beyond which the result is dominated by numerical error toward larger times. Comparing Figures 2 and 3, we can also see that even though the transients are quantitatively different, the two models have the same qualitative features: The amplitude of the variation of concentration about the threshold concentration at the source point decreases extremely rapidly, as does the width of the precipitation rings and gaps. In both simulations, we were able to clearly resolve two precipitation rings and two gaps, where the last gap is only visible by zooming in about five orders of magnitude. We cannot determine whether there is a third distinct ring; simulating this numerically would require at least one order of magnitude more resolution in space, due to the additional timestepping at least two orders of magnitude more in computational expense. In the following, we prove, for the simplified model, that the ring structure must break down within a finite interval; the simulations suggest that this interval is not particularly large.
We have also observed that most of the quantitative change comes from simplification (b). Implementing simplification (a) without simplification (b) results in visually very close, though not identical behavior to the full model.
Note that simplification (a) implies that there is no precipitation below the line x 2 = α 2 t, even when u > u * . The advantage of this simplification is that onset of precipitation now ceases to be a free boundary problem and follows parabolic scaling. A motivation for the validity of this simplification comes from the following fact: it is proved in [9] that if the solution to the full HHMO-model converges to a parabolically self-similar profile as t → ∞, then the contribution to the HHMO-dynamics from precipitation below the parabola α 2 t = x 2 is asymptotically negligible.
Simplification (b) is justified by the numerical observation that the equation without the damping term pw already converges to the same profile, so that an additional linear damping toward the equilibrium will not make a qualitative difference. Moreover, assuming that the HHMO-solution converges to equilibrium, pw becomes asymptotically small while the right hand side of (10a) remains an order-one quantity.
We note, however, that it is very difficult to estimate the quantitative effect of (a) and (b) due to the discontinuous reaction term and the free boundary of onset of precipitation, so that a rigorous justification of these two steps remains open.
To proceed, we extend the simplified HHMO-model (11) to the entire real line by even reflection and abbreviate Proceeding formally, we apply the Duhamel principle-a detailed justification of the Duhamel principle in context of weak solutions is given in [6]-then change the order of integration and implement the change of variables s = y 2 /ζ 2 , so that where Θ is the standard heat kernel We are specifically interested in the solution on the parabola x 2 = α 2 t. For notational convenience, we assume in the following that x is nonnegative; solutions for negative x are obtained by even reflection. Then, setting ω(x) = w(x, x 2 /α 2 ) where Inserting the explicit expression for ρ into (15) and noting that ω is extended to negative arguments by even reflection, we obtain with and A graph of the kernel K is shown in Figure 4. In Appendix A, we give a combination of analytic and numerical evidence showing that this kernel satisfies properties (i)-(iii) stated in the introduction. We conclude that the simplified HHMO-model implies an integral equation of the form (1). Vice versa, given a solution ω to (17), we can reconstruct a solution to the PDE-formulation of the simplified HHMO-model. Indeed, setting we can repeat the calculation leading to (15), which proves that W (x, α −2 x 2 ) = ω(x). Thus, W solves (13) so that it provides a mild solution to (11).

Non-degenerate breakdown of precipitation bands
In this section, we investigate the structure of solutions to the integral equation (1) for kernels K which satisfy assumptions (i)-(iii). Specifically, we seek solutions ω defined on a half-open interval [0, x * ) or on [0, ∞) which change sign at isolated points x i for i = 1, 2, . . . , ordered in increasing sequence. Setting x 0 = 0 and noting that precipitation must occur in a neighborhood of the origin if it sets in at all, the precipitation bands are the intervals (x i , x i+1 ) for even integers i ≥ 0. Hence, where we write I A to denote the indicator function of a set A. Thus, the onedimensional precipitation equation (1) takes the form with For x ≥ x n−1 , we also define the partial sums Thus, ω n (x) = ω(x) for x ∈ [x n−1 , x n ]. With this notation in place, we are able to define the notion of degenerate solution. (22) holds up to some finite x i ≡ x * and it is not possible to apply this formula on [x * , x * + ε) for any ε > 0; it is non-degenerate if ω possesses a finite or infinite sequence of isolated zeros {x i } and the solution can be continued in the sense of (22) to some right neighborhood of any of its zeros.
In the remainder of this section, we characterize non-degenerate solutions. We cannot exclude that a solution is degenerate, i.e., that it cannot be continued at all beyond an isolated root; in fact, Section 5 shows that kernels with degenerate solutions exist. We note that a degenerate solution provides an extreme scenario of a breakdown in which the solution reaches equilibrium in finite time. Thus, the main result of this section, Theorem 4 below, can be understood as saying that even when the solution is non-degenerate, it still fails to exist outside of a bounded interval. Lemma 2. Suppose K ∈ C([0, 1]) is non-negative, strictly positive somewhere, and K(θ) = o(θ) as θ → 0. Then a non-degenerate solution to (1) has an infinite number of precipitation rings.
Proof. A non-degenerate solution, by definition, is a solution that can be extended to the right in some neighborhood of any of its zeros. Now suppose there is a largest zero x n . Then ω is well-defined and equals ω n+1 on [x n , ∞). Now let x > x n and consider the limit x → ∞. When n is even, since ρ i+1 (x) < ρ i (x), a contradiction. When n is odd, once again a contradiction.
Let ω be a non-degenerate solution to (1) with an infinite number of precipitation rings. Then x 2n /x 2n+1 → 1 as n → ∞. Moreover, x 2n+1 − x 2n , the width of the nth precipitation ring, converges to zero.
Proof. When the sequence {x i } is bounded, the claim is obvious. Thus, assume that this sequence is unbounded. Since K is negative on (1 − ε, 1) for sufficiently small ε > 0, K is positive on this interval. As in the proof of Lemma 2, We can directly conclude that x 2n /x 2n+1 → 1 as n → ∞. Further, noting that K(1) = 0 and using the fundamental theorem of calculus, we obtain where, by the mean value theorem of integration, the last equality holds for some ζ n ∈ [x 2n /x 2n+1 , 1]. Since ζ n → 1, we conclude that x 2n+1 −x 2n → 0 as n → ∞.
Proof. We begin by recalling the second order mean value theorem, which states that for a twice continuously differentiable function f and nodes a < b < c there exists y ∈ [a, c] such that We apply this result to the partial sum function ω n with a = x n , b = x n+1 , and c = x ∈ (x n+1 , x n+2 ]. We note that ω n (x n ) = 0. Further, subtracting (24) from (22), we obtain, for x ∈ [x n+1 , x n+2 ], that so that, in particular, ω n (x n+1 ) = (−1) n ρ n (x n+1 ). Equation (29) then reads To estimate the right hand expression, we compute where By direct computation, F (z) = z 2 K (z). Since K is unimodal, this implies that F has an isolated maximum on [0, 1]. Now suppose that x ∈ (x n+1 , x n+2 ). We consider two separate cases. When n is even, for every y ≥ x n−1 there exists a unique odd index such that the sequence of f i is strictly increasing for i = 1, . . . , − 1 and is strictly decreasing for i = + 1, . . . , n − 1. Hence, where M is a strictly positive constant. Further, ω(x) < 0. Inserting these two estimates into (31), we obtain When n is odd, for every y ≥ x n−1 there exists a unique even index such that the sequence of f i is strictly increasing for i = 0, . . . , − 1 and is strictly decreasing for i = + 1, . . . , n − 1. Hence, ω n (y) = (f 0 − f 1 ) + · · · + f + (−f +1 + f +2 ) + · · · + (−f n−2 + f n−1 ) Further, ω(x) > 0. As before, inserting these two estimates into (31), we obtain Thus, we again obtain an estimate of exactly the form (35) and we do not need to further distinguish between n even or odd.
To proceed, we define so that, by assumption, R(θ) → 1 as θ → 1. Further, Changing variables to we write inequality (35) in the form and Observe that G(0) = 0, G (0) > 0, G (q) < 0 for all q > 0, and G(1) = 2 1+σ −3 < 0. Hence, there is a unique root q * ∈ (0, 1) such that G(q) < 0 for all q > q * . Now fix ε > 0, define and consider any even index j for which q j > q * + ε. Since (41) was derived under the assumption x ∈ (x j+1 , x j+2 ), or equivalently q ∈ (0, q j ), this inequality must hold for each tuple (r j , q * + ε). Now if there were an infinite set of indices for which q j > q * + ε, we could pass to the limit j → ∞ on the subsequence of such indices. Since K (θ) < 0 for θ ∈ (θ * , 1) and K(θ) ∼ k (1 − θ) σ as θ → 1, Lemma 3 is applicable and implies that r 2k = x 2k /x 2k+1 → 1. As for any fixed q, each of the S i (r, q) converges to zero as r → 1, we arrive at the contradiction G(q * + ε) > 0. Hence, lim sup k→∞ k even To extend this result to odd n, we note that for all large enough even n. This implies an even stricter bound on the right hand side of (41) when n is replaced by n + 1, so that (44) holds on the subsequence of odd integers as well.
Altogether, this proves that the sequence of internodal distances d n = x n+1 − x n is geometric, thus the x n have a finite limit.
Remark 5. Note that in the proof of Theorem 4, we only need a result which is weaker than the statement of Lemma 3, namely that r n → 1 for even n going to infinity Remark 6. Note that the argument yields an explicit upper bound for q n , namely lim sup where, as in the proof, q * is the unique positive root of G, which is defined in (42e).
Remark 7. It is possible to relax the unimodality condition in the statement of Theorem 4. In fact, it suffices that lim θ 1 K (θ) = −∞. Indeed, assume that K is defined on U K . Take any θ * ∈ (0, 1) such that K < 0 on [θ * , 1) ∩ U K . Changing variables in (24), we obtain When n is even and x ∈ (x n+1 , x n+2 ), the singularity of K and K at θ = 1 is separated from the domain of integration. We can therefore differentiate under the integral, so that When n is odd, then for every x ∈ (x n+1 , x x+2 ) there is an even index such that x −1 < xθ * ≤ x +1 (with the provision that x −1 = 0), and where F is as in (33). As in the proof of the theorem, F (z) = z 2 K (z) < 0 on [θ * , 1) so that M * = max z∈[0,1] F (z) is finite and Therefore, Hence, (35) and (37) continue to hold and the remainder of the proof proceeds as before.
Corollary 8. Suppose that K satisfies conditions (i)-(iii) stated in the introduction. Then there exists x * < ∞ so that the maximal interval of existence of a precipitation ring pattern in the sense of (22 Proof. When the solution is degenerate, then such x * exists by definition. Otherwise, property (i) and the positivity of the kernel on (0, 1) imply that Lemma 2 is applicable, i.e., there exist an infinite number of precipitation rings. Then, due to properties (ii) and (iii), Theorem 4 applies and asserts the existence of a finite accumulation point x * of the ring pattern.

Existence of degenerate solutions
In this section, we show that degenerate solutions exist. These are solutions to (1) which cannot be continued past a finite number of zeros. While we cannot settle this question for the concrete kernel introduced in Section 3, we construct a kernel K such that the solution cannot be continued in the sense of (1) past x 2 , the end point of the first precipitation gap.
Theorem 9. There exist a non-negative kernel K ∈ C([0, 1]) and a constant Γ such that the solution of the integral equation (1) is degenerate. Moreover, K is differentiable at θ = 0 and satisfies conditions (i) and (ii).
Remark 10. The proof starts from a kernel template which is then modified on a subinterval [0, r) to produce a kernel K with the desired properties. When the kernel template is C 1 ([0, 1)) and C 2 ((0, 1)), the resulting kernel K will inherit these properties except possibly at the gluing point θ = r where continuity of the first derivative is not enforced. This is a matter of convenience, not of principle: The existence of degenerate solutions does not hinge on the existence of a jump discontinuity for K . Straightforward, yet technical modifications of the gluing construction employed in the proof will yield a kernel producing degenerate solutions within the same class of kernels to which Theorem 4, in the sense of Remark 7, applies.
Remark 11. In Section 3, we derived a concrete kernel by simplifying the HHMOmodel. In that setting, there exists an integrable function G such that K(θ) = θ 2 G(θ) and In the proof of the theorem, we preserve this relationship, i.e., the constant Γ here will also satisfy (53).

Extended solutions for the simplified HHMO-model
So far, we have seen that precipitation band patterns as a sequence of intervals in which ω(x) > 0, i.e. the reactant concentration exceeds the super-saturation threshold, must break down at a finite location x * which is either an accumulation point given by Theorem 4, or until a point at which the solution degenerates after a finite number of precipitation bands as in Theorem 9. In this section, we consider a more general notion of solution which is motivated by the construction of weak solutions to the full HHMO-model in [15].
Proof. Changing variables, we write (76) as Now consider a family of mollified Heaviside functions H ε ∈ C ∞ (R, [0, 1]) parameterized by ε > 0 such that H ε (z) = 1 for z ≥ ε and H ε (z) = 0 for z ≤ −ε. We claim that, for fixed ε > 0, the corresponding mollified equation has a solution ω ε ∈ C([0, ∞)). Indeed, suppose that ω ε is already defined on some interval [0, a], where a may be zero. We seek ω ε ∈ C([a, a + δ]) which continuously extends ω ε past x = a as a fixed point of a map T from C([a, a + δ]) endowed with the supremum norm into itself, defined by Since T is a strict contraction for δ > 0 small enough, hence has a unique fixed point. In addition, the maximal interval of existence of ω ε is closed, as the right hand side of (79) is continuous, and open at the same time due to the preceding argument. Thus, a solution ω ε ∈ C([0, ∞)) exists (and is unique). By direct inspection, for every fixed b > 0, the families {ω ε } and {H ε • ω ε } are uniformly bounded in C([0, b]) endowed with the supremum norm. Moreover, {ω ε } is equicontinuous. Indeed, for y, z ∈ (0, b], where, by the dominated convergence theorem, the right hand side converges to zero as z → y. Equicontinuity at y = 0 is obvious. Thus, by the Arzelá-Ascoli theorem, there exist a decreasing sequence ε i → 0 and a function ω ∈ C([0, b]) such that ω εi → ω in C([0, b]). Further, by the Banach-Alaoglu theorem, there exists ρ ∈ L ∞ ([0, b]) such that, possibly passing to a subsequence, H εi • ω εi ρ weakly- * in L ∞ . This implies that ρ takes values a.e. from the interval [0, 1] which contains the convex hull of the sequence. Passing to the limit in (79), we conclude that Finally, we claim that ρ(y) = 1 whenever ω(y) > 0. Indeed, fixing y such that ω(y) > 0, equicontinuity of ω ε implies that there exists a neighborhood of y on which ω εi is eventually strictly positive. On this neighborhood, H εi • ω εi converges strongly to 1. A similar argument proves that ρ(y) = 0 whenever ω(y) < 0. So far, we have shown that (ω, ρ) satisfy (76) and (77)  Uniqueness of extended solutions is a much more delicate issue. In the following particular case, we can give a positive answer to the question of uniqueness. for x ∈ [0, x * + ε]. Due to properties (i)-(iii) of the kernel, the problem falls into the general class of weakly degenerate cordial Volterra integral equations. In [8], we answer this question in the affirmative. While we believe that extended solutions are generically regularly extended, we cannot exclude the possibility that extended solutions develop a precipitation band pattern that accumulates at x * from above. Thus, the general question of unique extendability remains open.
We conjecture that the question of uniqueness of extended solutions might be addressed by replacing Definition 12 by a formulation in terms of a mixed linear complementarity problem. To be concrete, write ω = ω + − ω − , where ω + is the positive part and ω − the negative part of ω. Further, set σ = 1 − ρ and define the vector functions Then we can formulate the extended solution as follows. Find V ≥ 0, W ≥ 0 such that LV + MW + B = 0 (86a) subject to V, W = 0 , where L and M are linear operators defined by and The angle brackets in (86b) denote the canonical inner product for vectors of L 2 ((0, a))-functions, This formulation is known as a mixed linear complementarity problem. To make progress here, it is necessary to adapt Lions-Stampaccia theory [19] to the case of mixed complementarity problems; see, e.g., [4,27], for different reformulations in a Sobolev space setting.

Discussion
In this paper, we have identified a mechanism which leads to very rapid, i.e., finite-time equilibrization of a dynamical system with memory that is "damped" via relay hysteresis. Past a certain point, a solution can only be continued in a generalized sense by "completing the relay". We can assert existence and conditional uniqueness for the generalized solution, but full well-posedness remains open. Possible approaches are a reformulation of the concept of generalized solution in terms of a mixed linear complementarity problem as outlined above, or possibly a fixed point formulation using fractional integral operators. We believe that the integral equation (1) is a useful test bed for studying such approaches, with the hope to eventually transfer results to more general reaction-diffusion equations with relay hysteresis.
The detailed observed behavior is very much tied to property (i), the square-root degeneracy of the kernel K near θ = 1. From the perspective of solving integral equations, this behavior is too degenerate for classical contraction mapping arguments as used by Volterra to apply, but it is sufficiently non-degenerate that strong results can still be proved, see the discussion in [8]. This degeneracy is associated with the scaling behavior of the heat kernel, so even when an exact reduction from a PDE to an integral equation, as is possible for the simplified HHMO-model, is not available, the associated phenomenology is expected to survive.
Let us finally remark on the connection of our models to the real-world Liesegang precipitation phenomenon. Our numerics and also the detailed results on the breakdown of patterns for the simplified model should make it clear that we cannot expect that the pattern of rings seen in these models is a good literal description of the behavior of observed Liesegang rings. However, the fact that, independent of the details of simplified vs. full dynamics, the models converge extremely rapidly toward a steady-state which only exists as a generalized solution, we believe that it might be possible to interpret the fractional value of the precipitation function as a precipitation density. In this view, the fast reaction limit would provide a coarsegrained description of the phenomenon in the sense that the precise information about the location of the rings is lost, but the local average fraction of space covered by precipitation rings can still be asserted. This issue is a question about the validity of the fast reaction limit itself, which is beyond the scope of this paper. However, should the suggested interpretation be valid, the explicit form asymptotic profile detailed in Section 3 would establish a useful direct relationship between the parameters of the system and the precipitation density.
Furthermore, we note that the same lemma implies