{
  "id": "1904.10129",
  "version": "v1",
  "published": "2019-04-23T03:04:14.000Z",
  "updated": "2019-04-23T03:04:14.000Z",
  "title": "Decay in the one dimensional generalized Improved Boussinesq equation",
  "authors": [
    "Christopher Maulén",
    "Claudio Muñoz"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space $H^1\\times H^2$, existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity $p$ is sufficiently large. In this paper we remove that condition on the power $p$ and prove decay to zero in terms of the energy space norm $L^2\\times H^1$, for any $p>1$, in two almost complementary regimes: (i) outside the light cone for all small, bounded in time $H^1\\times H^2$ solutions, and (ii) decay on compact sets of arbitrarily large bounded in time $H^1\\times H^2$ solutions. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-23T03:04:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boussinesq equation",
      "ill-posed shallow water waves model",
      "decay problem",
      "dimensional",
      "power type nonlinearity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}