Stability of Front Solutions of the Bidomain Allen–Cahn Equation on an Infinite Strip

Abstract

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we study the bidomain Allen–Cahn equation, in which the Laplacian of the classical Allen–Cahn equation is replaced by the bidomain operator, a Fourier multiplier operator whose symbol is given by a homogeneous rational function of degree two. The bidomain Allen–Cahn equation supports planar front solutions much like the classical case. In contrast to the classical case, however, these fronts are not necessarily stable due to a lack of maximum principle; they can indeed become unstable depending on the parameters of the system. In this paper, we prove nonlinear stability and instability results for bidomain Allen–Cahn fronts on an infinite two-dimensional strip. We show that previously established spectral stability/instability results in imply stability/instability in the space of bounded uniformly continuous functions by establishing suitable decay estimates of the resolvent kernel of the linearized operator.