{
  "id": "0902.0257",
  "version": "v1",
  "published": "2009-02-02T11:51:01.000Z",
  "updated": "2009-02-02T11:51:01.000Z",
  "title": "On global solutions and blow-up for Kuramoto-Sivashinsky-type models, and well-posed Burnett equations",
  "authors": [
    "V. A. Galaktionov",
    "E. Mitidieri",
    "S. I. Pohozaev"
  ],
  "comment": "34 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global existence and blow-up as well as uniform bounds are reviewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry's version of weighted Gronwall's inequalities, and (vi) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirichlet and \"Navier\" boundary conditions. For some related higher-order PDEs in N dimensions uniform boundedness of global solutions of the Cauchy problem are established. As another related application, the well-posed Burnett-type equations, which are a higher-order extension of the classic Navier-Stokes equations, are studied. As a simple illustration, a generalization of the famous Leray-Prodi-Serrin-Ladyzhenskaya regularity results is obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-02T11:51:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55"
    ],
    "keywords": [
      "global solutions",
      "well-posed burnett equations",
      "kuramoto-sivashinsky-type models",
      "semilinear parabolic partial differential equations",
      "th-order semilinear parabolic partial differential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.0257G"
    }
  }
}