We prove well-posedness (existence and uniqueness) results for a class of degenerate reaction-difiusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The existence result, which constitutes the main thrust of this paper, is proved by means of a nondegenerate approximation system, the Faedo-Galerkin method, and the compactness method. 1. Introduction. Our point of departure is a widely accepted model, the so-called bidomain model, for describing the cardiac electric activity in a physical domain › ‰ R 3 (the cardiac muscle) over a time span (0;T), T > 0. In this model the cardiac muscle is viewed as two superimposed (anisotropic) continuous media, referred to as the intracellular (i) and extracellular (e), which occupy the same volume and are separated from each other by the cell membrane. To state the model, we let ui = ui(t;x) and ue = ue(t;x) represent the spatial cellular at time t 2 (0;T) and location x 2 › of the intracellular and extracellular electric potentials, respectively. The difierence v = v(t;x) = uiiue is known as the transmembrane potential. The anisotropic properties of the two media are modeled by conductivity tensors Mi(t;x) and Me(t;x). The surface capacitance of the membrane is represented by a constant cm > 0. The transmembrane ionic current is represented by a nonlinear (cubic polynomial) function h(t;x;v) depending on time t, location x, and the value of the potential v. The stimulation currents applied to the intra- and extracellular space are represented by a function Iapp = Iapp(t;x). A prototype system that governs the cardiac electric activity is the following degenerate reaction-difiusion system (known as the bidomain equations) cm@tv i div(Mi(t;x)rui) + h(t;x;v) = Iapp; (t;x) 2 QT; cm@tv + div(Me(t;x)rue) + h(t;x;v) = Iapp; (t;x) 2 QT; (1) where QT denotes the time-space cylinder (0;T) £ ›. We complete the bidomain system (1) with Dirichlet boundary conditions for both the intra- and extracellular
['Mostafa Bendahmane', 'Kenneth H. Karlsen']
ANALYSIS OF A CLASS OF DEGENERATE REACTION-DIFFUSION SYSTEMS AND THE BIDOMAIN MODEL OF CARDIAC TISSUE