{
  "id": "1703.02134",
  "version": "v1",
  "published": "2017-03-06T22:31:33.000Z",
  "updated": "2017-03-06T22:31:33.000Z",
  "title": "Global behaviour of radially symmetric solutions stable at infinity for gradient systems",
  "authors": [
    "Emmanuel Risler"
  ],
  "comment": "52 pages, 14 figures. arXiv admin note: substantial text overlap with arXiv:1703.01221. text overlap with arXiv:1604.02002",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with radially symmetric solutions of systems of the form \\[ u_t = -\\nabla V(u) + \\Delta_x u \\] where space variable $x$ and and state-parameter $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of solutions \"stable at infinity\", that is approaching a spatially homogeneous equilibrium when $|x|$ approaches $+\\infty$, is investigated. It is proved that every such solutions approaches a stacked family of radially symmetric bistable fronts travelling to infinity. This behaviour is similar to the one of bistable solutions for gradient systems in one unbounded spatial dimension, described in a companion paper. It is expected (but unfortunately not proved at this stage) that behind these travelling fronts the solution again behaves as in the one-dimensional case (that is, the time derivative approaches zero and the solution approaches a pattern of stationary solutions).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-06T22:31:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35B40"
    ],
    "keywords": [
      "radially symmetric solutions stable",
      "gradient systems",
      "global behaviour",
      "symmetric bistable fronts travelling",
      "time derivative approaches zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}