{
  "id": "math/0408332",
  "version": "v1",
  "published": "2004-08-24T14:24:32.000Z",
  "updated": "2004-08-24T14:24:32.000Z",
  "title": "Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions",
  "authors": [
    "Ross Pinsky"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\\in R^n\\times(0,\\infty); &u(x,0) =g(x)\\ge0, x\\in R^n; &u\\ge0, $$ where $$ L=\\sum_{i,j=1}^na_{i,j}(x)\\frac{\\partial^2}{\\partial x_i \\partial x_j}+\\sum_{i=1}^nb_i(x)\\frac\\partial{\\partial x_i} $$ is a non-degenerate elliptic operator, $g\\in C(R^n)$ and the reaction term $f$ converges to $-\\infty$ at a super-linear rate as $u\\to\\infty$. We give a sharp minimal growth condition on $f$, independent of $L$, in order that there exist a universal, a priori upper bound for all solutions to the above Cauchy problem--that is, in order that there exist a finite function $M(x,t)$ on $R^n\\times(0,\\infty)$ such that $u(x,t)\\le M(x,t)$, for all solutions to the Cauchy problem. Assuming now in addition that $f(x,0)=0$, so that $u\\equiv0$ is a solution to the Cauchy problem, we show that under a similar growth condition, an intimate relationship exists between two seemingly disparate phenomena--namely, uniqueness for the Cauchy problem with initial data $g=0$ and the nonexistence of unbounded, stationary solutions to the corresponding elliptic problem. We also give a generic condition for nonexistence of nontrivial stationary solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-08-24T14:24:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K15",
      "35K55"
    ],
    "keywords": [
      "cauchy problem",
      "stationary solutions",
      "super-linear absorption",
      "universal bounds",
      "uniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8332P"
    }
  }
}