{
  "id": "1610.00821",
  "version": "v1",
  "published": "2016-10-04T02:07:02.000Z",
  "updated": "2016-10-04T02:07:02.000Z",
  "title": "Finite-time degeneration of hyperbolicity without blowup for quasilinear wave equations",
  "authors": [
    "Jared Speck"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \\partial_t^2 \\Psi + (1 + \\Psi)^P \\Delta \\Psi = 0$ for $P \\in \\lbrace 1,2 \\rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy formation that has not previously been studied for quasilinear wave equations. Specifically, using only energy methods and ODE-type techniques, we exhibit an open (in an appropriate Sobolev topology) set of data such that $\\Psi$ is initially near $0$ while $1 + \\Psi$ vanishes in finite time. In fact, generic data profiles, when appropriately rescaled, lead to the vanishing of $1 + \\Psi$ in finite time. The solution remains regular up to the degeneracy in the following sense: there is a high-order energy, featuring degenerate weights only at the top order, that remains bounded up to the time of first vanishing. When $P=1$, we show that any $C^1$ extension of $\\Psi$ to the future of a point where $1 + \\Psi = 0$ must exit the regime of hyperbolicity. Moreover, the Kretschmann scalar (which is a curvature invariant) of the Lorentzian metric corresponding to the wave equation blows up at those points. In particular, our results show that curvature blowup for the metric of a quasilinear wave equation does not always coincide with singularity formation in the solution variable. Similar phenomena occur when $P=2$, but in this case, the vanishing of $1 + \\Psi$ corresponds only to a breakdown in the strict hyperbolicity of the equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-04T02:07:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L80",
      "35L05",
      "35L72"
    ],
    "keywords": [
      "quasilinear wave equation",
      "finite-time degeneration",
      "finite time",
      "finite-time tricomi-type degeneracy formation",
      "generic data profiles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}