Electron. J. Diff. Equ., Vol. 2011(2011), No. 03, pp. 1-26.

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

Youssef Akdim, Jaouad Bennouna, Mounir Mekkour

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 4, 2011.
Math Subject Classifications: A7A15, A6A32, 47D20.
Key Words: Weighted Sobolev spaces; truncations; time-regularization; renormalized solutions.

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Youssef Akdim
Département de Mathématiques
Faculté des Sciences Dhar-Mahraz, Fès, Morocco
email: akdimyoussef@yahoo.fr
Jaouad Bennouna
Département de Mathématiques
Faculté des Sciences Dhar-Mahraz, Fès, Morocco
email: jbennouna@hotmail.com
Mounir Mekkour
Département de Mathématiques
Faculté des Sciences Dhar-Mahraz, Fès, Morocco
email: mekkour.mounir@yahoo.fr

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