{
  "id": "1609.05428",
  "version": "v1",
  "published": "2016-09-18T06:17:43.000Z",
  "updated": "2016-09-18T06:17:43.000Z",
  "title": "Bounds for the extremal parameter of nonlinear eigenvalue problems and application to the explosion problem in a flow",
  "authors": [
    "Asadollah Aghajani",
    "Alireza M. Tehrani"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the nonlinear eigenvalue problem $ L u = \\lambda f(u) $, posed in a smooth bounded domain $ \\Omega \\subseteq \\Bbb{R}^{N} $ with Dirichlet boundary condition, where $ L $ is a uniformly elliptic second-order linear differential operator, $ \\lambda > 0 $ and $ f:[0,a_{f}) \\rightarrow \\Bbb{R}_{+} $ $ (0 < a_{f} \\leqslant \\infty)$ is a smooth, increasing and convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $. First we present some upper and lower bounds for the extremal parameter $ \\lambda^{*} $ and the extremal solution $ u^{*} $. Then we apply the results to the operator $ L_A = - \\Delta + A c(x) $ with $ A>0 $ and $ c(x) $ is a divergence-free flow in $ \\Omega $. We show that, if $\\psi_{A,\\Omega}$ is the maximum of the solution $\\psi_{A}(x)$ of the equation $ L_A u = 1 $ in $\\Omega$ with Dirichlet boundary condition, then for any incompressible flow $ c(x) $ we have, $\\psi_{A,\\Omega} \\longrightarrow 0$ as $A \\longrightarrow \\infty$ if and only if $c(x)$ has no non-zero first integrals in $H_{0}^{1}(\\Omega)$. Also, taking $ c(x)=-x\\rho(|x|) $ where $\\rho$ is a smooth real function on $[0,1]$ then $c(x)$ is never divergence-free in unit ball $ B\\subset \\Bbb{R}^{N} $, but our results completely determine the behaviour of the extremal parameter $ \\lambda^{*}_{A} $ as $ A \\longrightarrow \\infty $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-18T06:17:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35P30",
      "35J91",
      "35B32"
    ],
    "keywords": [
      "nonlinear eigenvalue problem",
      "extremal parameter",
      "explosion problem",
      "dirichlet boundary condition",
      "elliptic second-order linear differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}