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

In this article, we study the problem $$displaylines{ frac{partial}{partial t} b(x, u)-hbox{div}(a(x,t,u,D u)) +H(x,t,u,Du) = fquad hbox{in } Omegaimes ]0,T[,cr b(x,u)(t=0)=b(x,u_0)quadhbox{in } Omega,cr u=0quadhbox{in } partialOmegaimes ]0,T[ }$$ 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 $b(x,u_0)in L^1(Omega)$.