{
  "id": "1502.05963",
  "version": "v1",
  "published": "2015-02-20T18:21:26.000Z",
  "updated": "2015-02-20T18:21:26.000Z",
  "title": "Two-end solutions to the Allen-Cahn equation in $\\mathbb{R}^{3}$",
  "authors": [
    "Changfeng Gui",
    "Yong Liu",
    "Juncheng Wei"
  ],
  "comment": "55 pages; comments welcome",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "In this paper we consider the Allen-Cahn equation $$ -\\Delta u = u-u^3 \\ \\mbox{in} \\ {\\mathbb R}^3 $$ We prove that for each $k\\in\\left( \\sqrt{2},+\\infty\\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$ \\| u-H(\\cdot -k \\ln r + c_k) \\|_{L^\\infty} \\to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-20T18:21:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "allen-cahn equation",
      "two-end solutions",
      "moduli space theory",
      "growth rate",
      "formal dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}