{
  "id": "1510.01971",
  "version": "v1",
  "published": "2015-10-07T14:51:57.000Z",
  "updated": "2015-10-07T14:51:57.000Z",
  "title": "Concentration Phenomenon in Some Non-Local Equation",
  "authors": [
    "Olivier Bonnefon",
    "Jérôme Coville",
    "Guillaume Legendre"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation $$\\partial\\_t u(t, x) = \\left(a(x) -- \\int\\_{\\Omega} k(x, y)u(t, y) dy\\right ) u(t, x) + \\int\\_{\\Omega} m(x, y)[u(t, y) -- u(t, x)] dy\\quad \\text{ for}\\quad (t, x) $\\in$ \\mathbb{R}\\_{+} \\times \\Omega,$$ together with the initial condition $u(0, \\cdot) = u0 \\quad \\text{ in }\\quad \\Omega$. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values $\\Omega$ is a bounded subset of $\\mathbb{R}^N$ , the kernels $k$ and $m$ respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait, we construct a positive stationary solution which belongs to the space of Radon measures on $\\Omega$. Moreover, when this '' stationary '' measure is regular and bounded, we prove its uniqueness and show that, for any non negative initial datum in $L^{\\infty} (\\Omega) \\cap L^1 (\\Omega)$, the solution of the Cauchy problem converges to this limit measure in $L^2 (\\Omega)$. We also construct an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. These numerical simulations seem to reveal some dependence of the limit measure with respect to the initial datum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-07T14:51:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-local equation",
      "concentration phenomenon",
      "long time behaviour",
      "limit measure",
      "population dynamics models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}