Front Dynamics in Non - Smooth Ignition Systems

We consider a non-smooth system which models front motion in a noisy, excitable media. Standard methods from matched asymptotics permit the construction of a family of traveling front solutions. Similar constructions have been carried out in [5] and [2], where travelling wave solutions of non-smooth systems with piecewise linear but discontinuous nonlinearity with fixed jump are constructed for models arising in mechanics and neurobiology. The model considered here arises from a consideration of membrane hydration in PEM fuel cells in which the protonic transport is approximated as a discontinuous function of membrane water content. Specifically, the membrane is “ignited” if the water content crosses a specific threshold, while it is “extinguished” when water content is below the threshold. We show that the pulse evolution fits naturally into the framework of the renormalization group methods developed to study the stability of slowly evolving patterns [4], [3]. We construct a family of smooth, monotone, composite solutions which describe the propagation of the ignited region across the fuel cell. In a noisy environment, the fronts lose monotonicity and the front position is unresolved because of multiple ignition points. In linearizing about the global manifold of the slowly evolving fronts, we address the challenge of choosing an appropriate Sobolev space for which the nonlinearity is Fréchet differentiable. Our renormalization group approach exploits the fact that the time dependence of the linearized operator is on a slower time scale than the decay. The main result describes the exponential decay of the remainder and after the decay of the initial transient perturbations, we recover the formal pulse velocities at leading order.