  We prove in this paper the existence of solutions of nonlinear parabolic
problems in inhomogeneous Musielak Orlicz Sobolev spaces, we assume neither a
$\Delta_2$ nor $\nabla_2$ on the Musielak function $\varphi$. The main
contribution of our work is to prove the existence of entropy solutions without
the sign condition on the nonlinearity. The second term $f$ belongs to
$L^1(Q)$.
