Convergence of front propagation for anisotropic bistable reaction–diffusion equations

Abstract

We study the convergence of the singularly perturbed anisotropic, nonhomogeneous reaction–diffusion equation ε∂_tu−ε²divT°(x,∇u)+f(u)−ε(c₁/c₀)g=0, where f is the derivative of a bistable quartic-like potential with unequal wells, T°(x,·) 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(ε³|log ε|²) error bound (in the Hausdorff distance) with respect to a hypersurface moving with the geometric law V=(c−εκ_ϕ)n_ϕ+g-dependent terms, where n_ϕ is the so-called Cahn–Hoffmann vector and κ_ϕ 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.