{
  "id": "math/0208111",
  "version": "v1",
  "published": "2002-08-14T08:18:13.000Z",
  "updated": "2002-08-14T08:18:13.000Z",
  "title": "On zero mass solutions of viscous conservation laws",
  "authors": [
    "Grzegorz Karch",
    "Maria E. Schonbek"
  ],
  "comment": "To appear in Comm. Partial Diff. Eq",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial conditions, the exponent q^* = 1+1/(n+beta) is critical in the following sense. For q > q^* the large time behavior in L^p(R^n), 1 <= p <= infinity, of solutions is described by self-similar solutions to the linear heat equation. For q = q^*, we prove that the convection-diffusion equation with f(u) = u|u|^{q^*-1} has a family of self-similar solutions which play an important role in the large time asymptotics of general solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-08-14T08:18:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35K55"
    ],
    "keywords": [
      "zero mass solutions",
      "viscous conservation laws",
      "large time behavior",
      "self-similar solutions",
      "initial condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......8111K"
    }
  }
}