Electronic Journal of Differential Equations,
Vol. 2011(2011), No. 03, pp. 1-26.

Title: Solvability of degenerated parabolic equations without sign
       condition and three unbounded nonlinearities

Authors: Youssef Akdim   (Faculte des Sciences Dhar-Mahraz, Morocco)
         Jaouad Bennouna (Faculte des Sciences Dhar-Mahraz, Morocco)
	 Mounir Mekkour  (Faculte des Sciences Dhar-Mahraz, Morocco)

Abstract:
 In this article, we study the problem
 $$\displaylines{
 \frac{\partial}{\partial t} b(x, u)-\hbox{div}(a(x,t,u,D u))
 +H(x,t,u,Du) = f\quad \hbox{in }  \Omega\times ]0,T[,\cr
 b(x,u)(t=0)=b(x,u_0)\quad\hbox{in } \Omega,\cr
 u=0\quad\hbox{in } \partial\Omega\times ]0,T[
 }$$
 in the framework of weighted Sobolev spaces, with $b(x,u)$
 unbounded function on u. The main contribution of our work is to
 prove the existence of a  renormalized solution without the sign
 condition and the coercivity condition on $H(x,t,u,Du)$. The
 critical growth condition on $H$ is with respect to
 Du and no growth condition with respect to u.
 The  second term f belongs to  $L^1(Q)$, and
 $b(x,u_0)\in L^1(\Omega)$. 

Submitted June 28, 2010. Published January 04, 2011.
Math Subject Classifications: A7A15, A6A32, 47D20.
Key Words: Weighted Sobolev spaces; truncations;
           time-regularization; renormalized solutions.