{
  "id": "1802.03717",
  "version": "v1",
  "published": "2018-02-11T09:42:37.000Z",
  "updated": "2018-02-11T09:42:37.000Z",
  "title": "Localized peaking regimes for quasilinear parabolic equations",
  "authors": [
    "Andrey E. Shishkov",
    "Evgeniya A. Evgenieva"
  ],
  "comment": "26 p",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with asymptotic behavior as $t\\rightarrow T<\\infty$ of all weak (energy) solutions of a class of equations with model representative: \\begin{equation*} (|u|^{p-1}u)_t-\\Delta_p(u)+b(t,x)|u|^{\\lambda-1}u=0 \\quad (t,x)\\in(0,T)\\times\\Omega,\\,\\Omega\\in\\mathbb{R}^n,\\,n>1, \\end{equation*} with prescribed global energy function \\begin{equation*} E(t):=\\int_{\\Omega}|u(t,x)|^{p+1}dx+ \\int_0^t\\int_{\\Omega}|\\nabla_xu(\\tau,x)|^{p+1}dxd\\tau \\rightarrow\\infty\\ \\text{ as }t\\rightarrow T. \\end{equation*} Here $\\Delta_p(u)=\\sum_{i=1}^n\\left(|\\nabla_xu|^{p-1}u_{x_i}\\right)_{x_i}$, $p>0$, $\\lambda>p$, $\\Omega$ --- bounded smooth domain, $b(t,x)\\geqslant0$. Particularly, in the case \\begin{equation*} E(t)\\leqslant F_\\mu(t)=\\exp\\left(\\omega(T-t)^{-\\frac1{p+\\mu}}\\right)\\quad\\forall\\,t<T,\\,\\mu>0,\\,\\omega>0, \\end{equation*} it is proved that solution $u$ remains uniformly bounded as $t\\rightarrow T$ in arbitrary subdomain $\\Omega_0\\subset\\Omega:\\overline{\\Omega}_0\\subset\\Omega$ and there is obtained some sharp upper estimate of $u(t,x)$ when $t\\rightarrow T$ in dependence of $\\mu>0$ and $s=dist(x,\\partial\\Omega)$. In the case $b(t,x)>0$ $\\forall\\,(t,x)\\in(0,T)\\times\\Omega$ there are found sharp sufficient conditions on degeneration of $b(t,x)$ near $t=T$, guaranteeing for arbitrary (even large) solution mentioned above boundedness, and obtained sharp upper estimate of final profile of solution when $t\\rightarrow T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-11T09:42:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K59",
      "35B44",
      "35K58",
      "35K65"
    ],
    "keywords": [
      "quasilinear parabolic equations",
      "localized peaking regimes",
      "sharp upper estimate",
      "sharp sufficient conditions",
      "prescribed global energy function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}