{
  "id": "1010.5461",
  "version": "v2",
  "published": "2010-10-26T17:02:09.000Z",
  "updated": "2011-01-25T10:58:25.000Z",
  "title": "Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations",
  "authors": [
    "María J. Cáceres",
    "José A. Cañizo",
    "Stéphane Mischler"
  ],
  "journal": "Journal de Math\\'emathiques Pures et Appliqu\\'ees, Vol. 96, No. 4, pp. 334-362 (2011)",
  "doi": "10.1016/j.matpur.2011.01.003",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the asymptotic behavior of linear evolution equations of the type \\partial_t g = Dg + Lg - \\lambda g, where L is the fragmentation operator, D is a differential operator, and {\\lambda} is the largest eigenvalue of the operator Dg + Lg. In the case Dg = -\\partial_x g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg = -x \\partial_x g, it is known that {\\lambda} = 2 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation \\partial_t f = Lf. By means of entropy-entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In both cases mentioned above we show these conditions are met for a wide range of fragmentation coefficients, so the exponential convergence holds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-01-25T10:58:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "growth-fragmentation equation",
      "linear evolution equation",
      "asymptotic profile",
      "case dg",
      "exponential convergence holds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.5461C"
    }
  }
}