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.