{
  "id": "1309.7441",
  "version": "v2",
  "published": "2013-09-28T08:44:04.000Z",
  "updated": "2014-06-18T07:54:29.000Z",
  "title": "Long time behavior of solutions of a reaction-diffusion equation on unbounded intervals with Robin boundary conditions",
  "authors": [
    "Xinfu Chen",
    "Bendong Lou",
    "Maolin Zhou",
    "Thomas Giletti"
  ],
  "comment": "27 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the long time behavior, as $t\\to\\infty$, of solutions of $$ \\left\\{ \\begin{array}{ll} u_t = u_{xx} + f(u), & x>0, \\ t >0,\\\\ u(0,t) = b u_x(0,t), & t>0,\\\\ u(x,0) = u_0 (x)\\geqslant 0 , & x\\geqslant 0, \\end{array} \\right. $$ where $b\\geqslant 0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\\{ \\sigma \\phi \\}_{\\sigma >0}$, where $\\phi$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value $\\sigma^*$ such that the solution converges uniformly to 0 for any $0 < \\sigma < \\sigma^*$, and locally uniformly to a positive stationary state for any $ \\sigma > \\sigma^*$. In the threshold case $\\sigma= \\sigma^*$, the profile of the solution approaches the symmetrically decreasing ground state with some shift, which may be either finite or infinite. In the latter case, the shift evolves as $C \\ln t$ where~$C$ is a positive constant we compute explicitly, so that the solution is traveling with a pulse-like shape albeit with an asymptotically zero speed. Depending on $b$, but also in some cases on the choice of the initial datum, we prove that one or both of the situations may happen.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-18T07:54:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35K15",
      "35B40"
    ],
    "keywords": [
      "long time behavior",
      "robin boundary conditions",
      "reaction-diffusion equation",
      "unbounded intervals",
      "initial datum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.7441C"
    }
  }
}