Stationary bistable fronts on a periodic necklace graph: existence, stability and multiplicity

Abstract

We study stationary fronts for a bistable reaction-diffusion equation with the McKean reaction on a periodic necklace metric graph. For profiles that agree on each pair of parallel edges, the problem reduces to a scalar equation on the real line with a periodic, piecewise constant diffusion coefficient. The diffusion contrast of the necklace is not small, so the coefficient cannot be treated as a small perturbation of a constant. Because the McKean reaction is piecewise linear, the equation is affine between threshold crossings and coefficient interfaces, and a front is determined explicitly by its crossing points. We find an explicit interval of reaction thresholds, the pinning interval, such that stationary fronts, monotone or not, exist exactly for thresholds in this interval. Up to translations by the spatial period, there are exactly two decreasing fronts at each interior threshold and one front at each endpoint. For each fixed interior threshold, one decreasing front is Lyapunov stable and the other is unstable in the $W^{1,\infty}(\mathbb{R})$ norm, among mild solutions with one continuously moving threshold crossing. The crossing phases of the two decreasing fronts also determine nonmonotone fronts. For each interior threshold and odd $m\geq3$, choose $m$ periods of the coefficient that are sufficiently far apart. There are then exactly $2^m$ fronts with one threshold crossing in each chosen period and none elsewhere. Varying the chosen periods gives infinitely many inequivalent fronts with $m$ crossings.