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

Akdim, Youssef ; Bennouna, Jaouad ; Mekkour, Mounir

Electronic Journal of Differential Equations, Tome 2011 (2011).

Résumé

Summary: 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 $b(x,u_0)\in L^1(\Omega)$.

Détail

Classification : 47D20
Keywords: weighted Sobolev spaces, truncations, time-regularization, renormalized solutions