Planar front Instabilities of the Bidomain Allen-Cahn Equation
Yochiro Mori (May 7, 2020)
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The bidomain model is the standard model describing electrical activity of the heart. We discuss the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allenâ€?Cahn equation) in two spatial dimensions. In the bidomain Allenâ€?Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allenâ€?Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allenâ€?Cahn equation in striking contrast to the classical or anisotropic Allenâ€?Cahn equations. We identify two types of instabilities, one with respect to longâ€?wavelength perturbations, the other with respect to mediumâ€?wavelength perturbations. Interestingly, whether the front is stable or unstable under longâ€?wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediateâ€?wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediateâ€?wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions. Time permitting, I will also discuss properties of the bidomain FitzHugh Nagumo equations. This is joint work with Hiroshi Matano, Mitsunori Nara and Koya Sakakibara.