{
  "id": "1704.04938",
  "version": "v1",
  "published": "2017-04-17T11:45:00.000Z",
  "updated": "2017-04-17T11:45:00.000Z",
  "title": "A gradient flow generated by a nonlocal model of a neural field in an unbounded domain",
  "authors": [
    "Severino Horácio da Silva",
    "Antônio Luiz Pereira"
  ],
  "comment": "16 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we consider the non local evolution equation $$ \\frac{\\partial u(x,t)}{\\partial t} + u(x,t)= \\int_{\\mathbb{R}^{N}}J(x-y)f(u(y,t))\\rho(y)dy+ h(x). %\\,\\,\\, h \\geq 0. $$ We show that this equation defines a continuous flow in both the space $C_{b}(\\mathbb{R}^{N})$ of bounded continuous functions and the space $C_{\\rho}(\\mathbb{R}^{N})$ of continuous functions $u$ such that $u \\cdot \\rho$ is bounded, where $\\rho $ is a convenient \"weight function\"'. We show the existence of an absorbing ball for the flow in $C_{b}(\\mathbb{R}^{N})$ and the existence of a global compact attractor for the flow in $C_{\\rho}(\\mathbb{R}^{N})$, under additional conditions on the nonlinearity. We then exhibit a continuous Lyapunov function which is well defined in the whole phase space and continuous in the $C_{\\rho}(\\mathbb{R}^{N})$ topology, allowing the characterization of the attractor as the unstable set of the equilibrium point set. We also illustrate our result with a concrete example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-17T11:45:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "45J05",
      "37B25"
    ],
    "keywords": [
      "gradient flow",
      "nonlocal model",
      "neural field",
      "unbounded domain",
      "non local evolution equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}