STABILITY OF FRONT SOLUTIONS OF THE BIDOMAIN ALLEN--CAHN EQUATION ON AN INFINITE STRIP.

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 L² imply stability/instability in the space of bounded uniformly continuous functions by establishing suitable decay estimates of the resolvent kernel of the linearized operator. [ABSTRACT FROM AUTHOR]