{
  "id": "1206.0484",
  "version": "v1",
  "published": "2012-06-03T20:18:12.000Z",
  "updated": "2012-06-03T20:18:12.000Z",
  "title": "Slowly oscillating wavefronts of the KPP-Fisher delayed equation",
  "authors": [
    "Karel Hasik",
    "Sergei Trofimchuk"
  ],
  "comment": "25 pages, submitted",
  "journal": "Discrete and Continuous Dynamical Systems - A 34 (2014) 3511 - 3533",
  "doi": "10.3934/dcds.2014.34.3511",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\\phi(x \\nu +ct) >0,$ $ |\\nu|=1, $ satisfying $\\phi(-\\infty)=0$) to the delayed KPP-Fisher equation $$u_t(t,x) = \\Delta u(t,x) + u(t,x)(1-u(t-\\tau,x)), \\ u \\geq 0,\\ x \\in \\R^m. \\eqno(*)$$ First, we show that each semi-wavefront should be either monotone or slowly oscillating. Then a complete solution to the problem of existence of semi-wavefronts is provided. We prove next that the semi-wavefronts are in fact wavefronts (i.e. additionally $\\phi(+\\infty)=1$) if $c \\geq 2$ and $\\tau \\leq 1$; our proof uses dynamical properties of some auxiliary one-dimensional map with the negative Schwarzian. The analysis of the fronts' asymptotic expansions at infinity is another key ingredient of our approach. It allows to indicate the maximal domain ${\\mathcal D}_n$ of $(\\tau,c)$ where the existence of non-monotone wavefronts can be expected. Here we show that the problem of wavefront's existence is closely related to the Wright's global stability conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-03T20:18:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34K12",
      "35K57",
      "92D25"
    ],
    "keywords": [
      "kpp-fisher delayed equation",
      "slowly oscillating wavefronts",
      "semi-wavefront",
      "wrights global stability conjecture",
      "auxiliary one-dimensional map"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.0484H"
    }
  }
}