Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension

Abstract

The aim of this paper is to prove that, for specific initial data [Formula: see text] and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen–Cahn equation on the interval [Formula: see text] shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the “energy approach” proposed by Bronsard and Kohn [On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990) 983–997], if [Formula: see text] is the diffusion coefficient, we show that in a time scale of order [Formula: see text] nothing happens and the solution maintains the same number of transitions of its initial datum [Formula: see text]. The novelty consists mainly in the role of the initial velocity [Formula: see text], which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen–Cahn equation with relaxation.