{
  "id": "1604.04512",
  "version": "v1",
  "published": "2016-04-15T14:00:05.000Z",
  "updated": "2016-04-15T14:00:05.000Z",
  "title": "Large time behaviour of solutions to parabolic equations with Dirichlet operators and nonlinear dependence on measure data",
  "authors": [
    "Tomasz Klimsiak",
    "Andrzej Rozkosz"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study large time behaviour of solutions of the Cauchy problem for equations of the form $\\partial_tu-L u+\\lambda u=f(x,u)+g(x,u)\\cdot\\mu$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form ${\\mathcal{E}}$ and $\\mu$ is a nonnegative bounded smooth measure with respect to the capacity determined by ${\\mathcal{E}}$. We show that under the monotonicity and some integrability assumptions on $f,g$ as well as some assumptions on the form ${\\mathcal{E}}$, $u(t,x)\\rightarrow v(x)$ as $t\\rightarrow\\infty$ for quasi-every $x$, where $v$ is a solution of some elliptic equation associated with our parabolic equation. We also provide the rate convergence. Some examples illustrating the utility of our general results are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-15T14:00:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35K58",
      "60H30"
    ],
    "keywords": [
      "parabolic equation",
      "dirichlet operators",
      "nonlinear dependence",
      "measure data",
      "study large time behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160404512K"
    }
  }
}