Abstract
We prove well-posedness (existence and uniqueness) results for aclass of degenerate reaction-diffusion systems. A prototype system belongingto this class is provided by the bidomain model, which is frequently used tostudy and simulate electrophysiological waves in cardiac tissue. The existenceresult, which constitutes the main thrust of this paper, is proved by means ofa nondegenerate approximation system, the Faedo-Galerkin method, and thecompactness method.
Highlights
Our point of departure is a widely accepted model, the so-called bidomain model, for describing the cardiac electric activity in a physical domain Ω ⊂ R3 over a time span (0, T ), T > 0
We prove existence of weak solutions for the bidomain system (1) and the nonlinear system (4) using specific nondegenerate approximation systems
We prove existence of solutions to (6) by applying the Faedo-Galerkin method, deriving a priori estimates, and passing to the limit in the approximate solutions using monotonicity and compactness arguments
Summary
Our point of departure is a widely accepted model, the so-called bidomain model, for describing the cardiac electric activity in a physical domain Ω ⊂ R3 (the cardiac muscle) over a time span (0, T ), T > 0. Reaction-diffusion system, degenerate, weak solution, existence, uniqueness, bidomain model, cardiac electric field. If Mi ≡ λMe for some constant λ ∈ R, the system (1) is equivalent to a scalar parabolic equation for the transmembrane potential v This nondegenerate case, which assumes an equal anisotropic ratio for the intra- and extracellular media, is known as the monodomain model. Where I denotes the identity matrix, σlj and σtj, j = i, e, are the conductivity coefficients respectively along and across the cardiac fibers for the intracellular (j = i), extracellular (j = e) media, which are assumed to be the positive constants, while a = a(x) is the unit vector tangent to the fibers at a point x. The conductivity of the composite medium is characterized by M := Mi + Me
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