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

Y. Akdim¹ ; J. Bennouna² ; M. Mekkour² ; H. Redwane³

¹ Département de Mathématiques Laboratoire d'Analyse Mathématique et Applications Faculté des Sciences Dhar-Mahraz Fès, Morocco
² Département de Mathématiques Laboratoire d'Analyse Mathématiques et Applications Faculté des Sciences Dhar-Mahraz Fès, Morocco
³ Faculté des Sciences Juridiques, Économiques et Sociales Université Hassan 1 B.P. 784 Settat, Morocco

Applicationes Mathematicae, Tome 39 (2012) no. 1, pp. 1-22

Résumé

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 $b(x,u_0)\in L^1(\varOmega)$.

Détail

DOI : 10.4064/am39-1-1

Keywords: study problem begin gather* frac partial partial mathop div u u quad text varomega times quad text varomega quad text partial varomega times end gather* main contribution work prove existence renormalized solution without sign condition coercivity condition u critical growth condition only respect respect datum assumed varomega varomega