Well-posedness of a parabolic problem based on a bidomain model for electrophysiological wave propagation
Résumé
The purpose of the current paper is to study the well-posedness of the bidomain model. That model is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model describing two potentials that satisfy a system of coupled parabolic and elliptic PDEs, these being coupled with one or more ODEs representing the ionic activity. The parabolic and elliptic PDEs are first converted into a single parabolic PDE by the introduction of the so-called bidomain operator. We properly define and analyze that bidomain operator. We then present a proof of existence, uniqueness and regularity of a local (in time) solution through a semi-group approach. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch models.