{
  "id": "1501.05716",
  "version": "v1",
  "published": "2015-01-23T05:13:02.000Z",
  "updated": "2015-01-23T05:13:02.000Z",
  "title": "Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries",
  "authors": [
    "Hong Gu",
    "Bendong Lou",
    "Maolin Zhou"
  ],
  "comment": "40 pages, 6 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider Fisher-KPP equation with advection: $u_t=u_{xx}-\\beta u_x+f(u)$ for $x\\in (g(t),h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient $-\\beta$ on the long time behavior of the solutions. We find two parameters $c_0$ and $\\beta^*$ with $\\beta^*>c_0>0$ which play key roles in the dynamics, here $c_0$ is the minimal speed of the traveling waves of Fisher-KPP equation. More precisely, by studying a family of the initial data $\\{ \\sigma \\phi \\}_{\\sigma >0}$ (where $\\phi$ is some compactly supported positive function), we show that, (1) in case $\\beta\\in (0,c_0)$, there exists $\\sigma^*\\geqslant0$ such that spreading happens when $\\sigma > \\sigma^*$ and vanishing happens when $\\sigma \\in (0,\\sigma^*]$; (2) in case $\\beta\\in (c_0,\\beta^*)$, there exists $\\sigma^*>0$ such that virtual spreading happens when $\\sigma>\\sigma^*$ (i.e., $u(t,\\cdot;\\sigma \\phi)\\to 0$ locally uniformly in $[g(t),\\infty)$ and $u(t,\\cdot + ct;\\sigma \\phi )\\to 1$ locally uniformly in $\\R$ for some $c>\\beta -c_0$), vanishing happens when $\\sigma\\in (0,\\sigma^*)$, and in the transition case $\\sigma=\\sigma^*$, $u(t, \\cdot+o(t);\\sigma \\phi)\\to V^*(\\cdot-(\\beta-c_0)t )$ uniformly, the latter is a traveling wave with a \"big head\" near the free boundary $x=(\\beta-c_0)t$ and with an infinite long \"tail\" on the left; (3) in case $\\beta = c_0$, there exists $\\sigma^*>0$ such that virtual spreading happens when $\\sigma > \\sigma^*$ and $u(t,\\cdot;\\sigma \\phi)\\to 0$ uniformly in $[g(t),h(t)]$ when $\\sigma \\in (0,\\sigma^*]$; (4) in case $\\beta\\geqslant \\beta^*$, vanishing happens for any solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-23T05:13:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K20",
      "35K55",
      "35R35",
      "35B40"
    ],
    "keywords": [
      "long time behavior",
      "free boundary",
      "fisher-kpp equation",
      "virtual spreading happens",
      "vanishing happens"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150105716G"
    }
  }
}