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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 fora hyperbolic variation of Allen-Cahn equation on the interval $[a,b]$ sharesthe well-known dynamical metastability valid for the classical parabolic case.In particular, using the "energy approach" proposed by Bronsard and Kohn [8],if $\varepsilon\ll 1$ is the diffusion coefficient, we show that in a timescale of order $\varepsilon^{-k}$ nothing happens and the solution maintainsthe same number of transitions of its initial datum $u_0$. The novelty consistsmainly in the role of the initial velocity $u_1$, which may create or eliminatetransitions in later times. Numerical experiments are also provided in theparticular case of the Allen-Cahn equation with relaxation.

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