{
  "id": "1312.3431",
  "version": "v2",
  "published": "2013-12-12T10:12:23.000Z",
  "updated": "2014-02-19T11:42:19.000Z",
  "title": "The Behavior of the Free Boundary for Reaction-Diffusion Equations with Convection in an Exterior Domain with Neumann or Dirichlet Boundary Condition",
  "authors": [
    "Ross G. Pinsky"
  ],
  "comment": "The title of this new version of the paper is a little different. This version has some additional results. Also, some errors in the previous version were corrected",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let \\begin{equation*} L=\\sum_{i,j=1}^da_{i,j}\\frac{\\partial^2}{\\partial x_i\\partial x_j}-\\sum_{i=1}^db_i\\frac{\\partial}{\\partial x_i} \\end{equation*} be a second order elliptic operator and consider the reaction-diffusion equation with Neumann boundary condition, \\begin{equation*} \\begin{aligned} &Lu=\\Lambda u^p\\ \\text{in}\\ \\mathbb{R}^d-D;\\\\ &\\nabla u\\cdot \\bar n=-h\\ \\text{on}\\ \\partial D;\\\\ &u\\ge0 \\ \\text{is minimal}, \\end{aligned} \\end{equation*} where $p\\in(0,1)$, $d\\ge2$, $h$ and $\\Lambda$ are continuous positive functions, $D\\subset R^d$ is bounded, and $\\bar n$ is the unit inward normal to the domain $\\mathbb{R}^d-\\bar D$. Consider also the same equations with the Neumann boundary condition replaced by the Dirichlet boundary condition; namely, $u=h$ on $\\partial D$. The solutions to the above equations may possess a free boundary. When $D=\\{|x|<R\\}$ and $L$ and $\\Lambda$ are radially symmetric, we write the solution as $u(r)$ with $r=|x|$ and define the radius of the free boundary by $r^*(h)=\\inf\\{r>R:u(r)=0\\}$. We normalize the diffusion coefficient to be on unit order, consider the convection vector field to be on order $r^m$, $m\\in R$, pointing either inward $(-)$ or outward $(+)$, and consider the reaction coefficient $\\Lambda$ to be on order $r^{-j}$, $j\\in R$. For both the Neumann boundary case and the Dirichlet boundary case, we show for which choices of $m$, $(\\pm)$ and $j$ a free boundary exists, and when it exists, we obtain its growth rate in $h$ as a function of $m$, $(\\pm)$ and $j$. These results are then used to study the free boundary in the non-radially symmetric case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-19T11:42:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free boundary",
      "dirichlet boundary condition",
      "reaction-diffusion equation",
      "exterior domain",
      "neumann boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.3431P"
    }
  }
}