{
  "id": "2004.09102",
  "version": "v1",
  "published": "2020-04-20T07:35:52.000Z",
  "updated": "2020-04-20T07:35:52.000Z",
  "title": "Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel",
  "authors": [
    "Matthieu Alfaro",
    "Otared Kavian"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the semilinear diffusion equation $\\partial$ t u = Au + |u| $\\alpha$ u in the half-space R N + := R N --1 x (0, +$\\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ' , 0) = 0 for x ' $\\in$ R N --1. We prove that if the symbol of the operator A is of order a|$\\xi$| $\\beta$ near the origin $\\xi$ = 0, for some $\\beta$ $\\in$ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 < $\\alpha$ $\\le$ $\\beta$/(N + 1). On the other hand, we prove existence of positive global solutions of the semilinear diffusion equation in a half-space when $\\alpha$ > $\\beta$/(N + 1). Notice that in the case of the half-space, the exponent $\\beta$/(N + 1) is smaller than the so-called Fujita exponent $\\beta$/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-20T07:35:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive solution",
      "blow-up phenomena",
      "dispersion kernel",
      "half-space",
      "semilinear diffusion equation blows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}