Convergence of front propagation for anisotropic bistable reaction–diffusion equations

Article type: Research Article

Authors: Bellettini, Giovanni | Colli Franzone, Piero | Paolini, Maurizio

Affiliations: Dipartimento di Matematica Applicata U. Dini, Università di Pisa, 56126 Pisa, Italy, E-mail: [email protected] | Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy, E-mail: [email protected] | Dipartimento di Matematica, Università Cattolica del Sacro Cuore, 25121 Brescia, Italy, E-mail: [email protected]

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−εκϕ)nϕ+g-dependent terms, where nϕ is the so-called Cahn–Hoffmann vector and κϕ 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.

DOI: 10.3233/ASY-1997-153-406

Journal: Asymptotic Analysis, vol. 15, no. 3-4, pp. 325-358, 1997