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

Raffaele Folino

Department of Information Engineering, Computer Science and Mathematics, Università degli Studi dell’Aquila, Via Vetoio, Coppito, 67100 L’Aquila, Italy

E-mail Address: raffaele.folino@graduate.univaq.it

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https://doi.org/10.1142/S0219891617500011Cited by:8 (Source: Crossref)

Abstract

The aim of this paper is to prove that, for specific initial data $(u_{0}, u_{1})$ and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen–Cahn equation on the interval $[a, b]$ 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 $ε ≪ 1$ is the diffusion coefficient, we show that in a time scale of order $ε^{- k}$ nothing happens and the solution maintains the same number of transitions of its initial datum $u_{0}$ . The novelty consists mainly in the role of the initial velocity $u_{1}$ , 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.

Communicated by P. Marcati

Keywords:

AMSC: 35B25, 35K57, 35L20

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