Metastability for Hyperbolic Variations of Allen–Cahn Equation

Abstract

The Allen–Cahn equation is a parabolic reaction–diffusion equation that has been originally proposed in Allen and Cahn (Acta Metall 27:1085–1095, 1979, [1]) to describe the motion of antiphase boundaries in iron alloys. In general, reaction–diffusion equations of parabolic type undergo the same criticisms of the linear diffusion equation, mainly concerning lack of inertia and infinite speed of propagation of disturbances. To avoid these unphysical properties, many authors proposed hyperbolic variations of the classic reaction–diffusion equations. Here, we consider a hyperbolic variation of the Allen–Cahn equation and present some results contained in Folino (J Hyperbolic Differ Equ 14:1–26, 2017, [6]) and Folino et al. (Metastable dynamics for hyperbolic variations of the Allen–Cahn equation, 2016, [8]) concerning the metastable dynamics of solutions. We study the singular limit of the solutions as the diffusion coefficient $\varepsilon \rightarrow 0^+$ and show that the hyperbolic version shares the well-known dynamical metastability valid for the parabolic equation.