{
  "id": "1510.07168",
  "version": "v1",
  "published": "2015-10-24T17:36:53.000Z",
  "updated": "2015-10-24T17:36:53.000Z",
  "title": "Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension",
  "authors": [
    "Raffaele Folino"
  ],
  "categories": [
    "math.AP"
  ],
  "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 [5], if $\\varepsilon\\ll 1$ is the diffusion coefficient, we show that in a time scale of order $\\varepsilon^{-k}$ nothing happens and the solutions maintain the same number of transitions of its initial datum $u_0$. The novelty of our result 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-24T17:36:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "allen-cahn equation",
      "hyperbolic variation",
      "space dimension",
      "slow motion",
      "initial datum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}