{
  "id": "1609.03857",
  "version": "v1",
  "published": "2016-09-13T14:28:35.000Z",
  "updated": "2016-09-13T14:28:35.000Z",
  "title": "Non-Autonomous Forms and Invariance",
  "authors": [
    "Dominik Dier"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We generalize the Beurling--Deny--Ouhabaz criterion for parabolic evolution equations governed by forms to the non-autonomous, non-homogeneous and semilinear case. Let $V, H$ are Hilbert spaces such that $V$ is continuously and densely embedded in $H$ and let $\\mathcal{A}(t)\\colon V\\to V^\\prime$ be the operator associated with a bounded $H$-elliptic form $\\mathfrak{a}(t,.,.)\\colon V\\times V \\to \\mathbb{C}$ for all $t \\in [0,T]$. Suppose $\\mathcal{C} \\subset H$ is closed and convex and $P \\colon H \\to H$ the orthogonal projection onto $\\mathcal{C}$. Given $f \\in L^2(0,T;V')$ and $u_0\\in \\mathcal{C}$, we investigate whenever the solution of the non-autonomous evolutionary problem \\[ u'(t)+\\mathcal{A}(t)u(t)=f(t), \\quad u(0)=u_0, \\] remains in $\\mathcal{C}$ and show that this is the case if Pu(t) \\in V \\quad \\text{and} \\quad \\operatorname{Re} \\mathfrak{a}(t,Pu(t),u(t)-Pu(t)) \\ge \\operatorname{Re} \\langle f(t), u(t)-Pu(t) \\rangle for a.e.\\ $t \\in [0,T]$. Moreover, we examine necessity of this condition and apply this result to a semilinear problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-13T14:28:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K90",
      "35K58"
    ],
    "keywords": [
      "non-autonomous forms",
      "invariance",
      "parabolic evolution equations",
      "non-autonomous evolutionary problem",
      "semilinear problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}