{
  "id": "1903.01774",
  "version": "v1",
  "published": "2019-03-05T11:50:21.000Z",
  "updated": "2019-03-05T11:50:21.000Z",
  "title": "A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology",
  "authors": [
    "István Balázs",
    "Philipp Getto",
    "Gergely Röst"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We establish variants of existing results on existence, uniqueness and continuous dependence for a class of delay differential equations (DDE). We apply these to continue the analysis of a differential equation from cell biology with state-dependent delay, implicitly defined as the time when the solution of a nonlinear ODE, that depends on the state of the DDE, reaches a threshold. For this application, previous results are restricted to initial histories belonging to the so-called solution manifold. We here generalize the results to a set of nonnegative Lipschitz initial histories which is much larger than the solution manifold and moreover convex. Additionally, we show that the solutions define a semiflow that is continuous in the state-component in the $C([-h,0],\\R^2)$ topology, which is a variant of established differentiability of the semiflow in $C^1([-h,0],\\R^2)$. For an associated system we show invariance of convex and compact sets under the semiflow for finite time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-05T11:50:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "state-dependent delay",
      "cell biology",
      "lipschitz functions",
      "continuous semiflow",
      "solution manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}