Convergence of front propagation for anisotropic bistable reaction-diffusion equations

Maurizio Paolini, Giovanni Bellettini, Piero Colli Franzone

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15 Citations (Scopus)

Abstract

We study the convergence of the singularly perturbed anisotropic, nonhomogeneous reaction-diffusion equation $\epsilon \partial_t u - \epsilon^2\text{div} T^o(x, \nabla u) + f(u) - \epsilon \frac{c_1}{c_0} g = 0$ where f is the derivative of a bistable quartic-like potential with unequal wells, $T^o (x, \cdot)$ is a nonlinear monotone operator homogeneous of degree one and g is a given forcing term. More precisely we prove that an appropriate level set of the solution satisfies an $O (\epsilon^3 |\log\epsilon|^2)$ error bound (in the Hausdorff distance) with respect to a hypersurface moving with the geometric law $V = (c - \epsilon \kappa_\phi) n_\phi +$ g-dependent terms, where $n_\phi$ is the so-called Cahn-Hoffmann vector and $\kappa_\phi$ denotes the anisotropic mean curvature of the hypersurface. We also discuss the connection between the anisotropic reaction-diffusion equation and the bidomain model, which is described by a system of equations modeling the propagation of an electric stimulus in the cardiac tissue.
Original languageEnglish
Pages (from-to)325-358
Number of pages34
JournalAsymptotic Analysis
Publication statusPublished - 1997

Keywords

  • anisotropy
  • electrocardiology
  • reaction-diffusion equations
  • singular perturbation

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