{
  "id": "1706.03403",
  "version": "v1",
  "published": "2017-06-11T20:24:39.000Z",
  "updated": "2017-06-11T20:24:39.000Z",
  "title": "Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay",
  "authors": [
    "Sergei Trofimchuk",
    "Vitaly Volpert"
  ],
  "comment": "36 pages, 3 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We consider a delayed reaction-diffusion equation $u_t(t,x) = u_{xx}(t,x) + g(u(t,x), u(t-\\tau,x)) \\ (*)$ with bistable nonlinearity $g$. In difference with previous works, we do not assume the monotonicity of $g(u,v)$ with respect to $v$ that hinders application of the comparison techniques. For two different types of $v-$unimodal nonlinearity $g(u,v)$, we prove the existence of a maximal continuous family of bistable monotone wavefronts $u= \\phi(x+c(\\tau)t,\\tau)$. Our proof is based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions where Lyapunov-Schmidt reduction is realised in appropriate weighted spaces of $C^2$-smooth functions. This method requires a detailed analysis of associated linear differential Fredholm operators and their formal adjoints. Depending on type of unimodality (equivalently, on the sign of $c(\\tau)$), two different scenarios can be observed for our wave solutions of $(*)$: i) independently on the size of delay $\\tau$, each bistable wavefront is monotone; ii) wavefronts are monotone for moderate values of $\\tau$ and can oscillate for large $\\tau$. Our results are illustrated by two biological models and one `toy' example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-11T20:24:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34K12",
      "35K57",
      "92D25"
    ],
    "keywords": [
      "unimodal bistable reaction-diffusion equation",
      "monotone waves",
      "global continuation",
      "associated linear differential fredholm operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}