{
  "id": "1603.01100",
  "version": "v1",
  "published": "2016-03-02T10:01:59.000Z",
  "updated": "2016-03-02T10:01:59.000Z",
  "title": "On evolution equations governed by non-autonomous forms",
  "authors": [
    "EL-Mennaoui Omar",
    "Laasri Hafida"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We consider a linear non-autonomous evolutionary Cauchy problem \\begin{equation} \\dot{u} (t)+A(t)u(t)=f(t) \\hbox{ for }\\ \\hbox{a.e. t}\\in [0,T],\\quad u(0)=u_0, \\end{equation} where the operator $A(t)$ arises from a time depending sesquilinear form $a(t,.,.)$ on a Hilbert space $H$ with constant domain $V.$ Recently a result on $L^2$-maximal regularity in $H,$ i.e., for each given $f\\in L^2(0,T,H)$ and $u_0 \\in V$ the problem above has a unique solution $u\\in L^2(0,T,V)\\cap H^1(0,T,H),$ is proved in [10] under the assumption that $a$ is symmetric and of bounded variation. The aim of this paper is to prove that the solutions of an approximate non-autonomous Cauchy problem in which $a$ is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form. In particular, this provide an alternative proof of the result in [10] on $L^2$-maximal regularity in $H.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-02T10:01:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K90",
      "35K50",
      "35K45",
      "47D06"
    ],
    "keywords": [
      "evolution equations",
      "non-autonomous forms",
      "maximal regularity",
      "linear non-autonomous evolutionary cauchy problem",
      "bounded variation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160301100O"
    }
  }
}