{
  "id": "math/0504333",
  "version": "v1",
  "published": "2005-04-15T21:15:15.000Z",
  "updated": "2005-04-15T21:15:15.000Z",
  "title": "Sharp Transition Between Extinction and Propagation of Reaction",
  "authors": [
    "Andrej Zlatos"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the reaction-diffusion equation \\[ T_t = T_{xx} + f(T) \\] on $\\bbR$ with $T_0(x) \\equiv \\chi_{[-L,L]} (x)$ and $f(0)=f(1)=0$. In 1964 Kanel' proved that if $f$ is an ignition non-linearity, then $T\\to 0$ as $t\\to\\infty$ when $L<L_0$, and $T\\to 1$ when $L>L_1$. We answer the open question of relation of $L_0$ and $L_1$ by showing that $L_0=L_1$. We also determine the large time limit of $T$ in the critical case $L=L_0$, thus providing the phase portrait for the above PDE with respect to a 1-parameter family of initial data. Analogous results for combustion and bistable non-linearities are proved as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-04-15T21:15:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35K15"
    ],
    "keywords": [
      "sharp transition",
      "extinction",
      "propagation",
      "large time limit",
      "initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4333Z"
    }
  }
}