Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Tom 39 / 2012

Y. Akdim, J. Bennouna, M. Mekkour, H. Redwane Applicationes Mathematicae 39 (2012), 1-22 MSC: Primary 47A15; Secondary 46A32, 47D20. DOI: 10.4064/am39-1-1

Streszczenie

We study the problem \begin{gather*} \frac{\partial b(x, u)}{\partial t}-{\mathop{\rm div}}(a(x,t,u,D u))+H(x,t,u,Du) = \mu\quad \text{in } Q=\varOmega\times (0,T),\\ b(x,u)|_{t=0}=b(x,u_0)\quad\text{in } \varOmega,\\ u=0\quad \text{in } \partial\varOmega\times (0,T). \end{gather*} The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on $H(x,t,u,Du)$. The critical growth condition on $H$ is only with respect to $Du$ and not with respect to $u$. The datum $\mu$ is assumed to be in $L^1(Q)+L^{p'}(0,T;W^{-1, p'}(\varOmega))$ and $b(x,u_0)\in L^1(\varOmega)$.

Autorzy

  • Y. AkdimDépartement de Mathématiques
    Laboratoire d'Analyse Mathématique
    et Applications
    Faculté des Sciences Dhar-Mahraz
    Fès, Morocco
    e-mail
  • J. BennounaDépartement de Mathématiques
    Laboratoire d'Analyse Mathématiques et Applications
    Faculté des Sciences Dhar-Mahraz
    Fès, Morocco
    e-mail
  • M. MekkourDépartement de Mathématiques
    Laboratoire d'Analyse Mathématiques et Applications
    Faculté des Sciences Dhar-Mahraz
    Fès, Morocco
    e-mail
  • H. RedwaneFaculté des Sciences Juridiques,
    Économiques et Sociales
    Université Hassan 1
    B.P. 784
    Settat, Morocco
    e-mail

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