We prove the existence of solutions to nonlinear parabolic problems of the following type: ∂b(u)⁄ ∂t + A(u) = f + div(Θ(x;t; u)) in Q, u(x;t) = 0 on ∂Ω × [0; T ], b(u)(t = 0) = b(u0) on Ω, where b : ℝ → ℝ is a strictly increasing function of class C 1 , the term A(u) = −div (a(x, t, u, ∇u)) is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, Θ: Ω × [0; T ] × ℝ → ℝ is a Carathéodory, noncoercive function which satisfies the following condition: sup |s|≤k |Θ(·, ·, s)| ∈ Eψ(Q) for all k > 0, where ψ is the Musielak complementary function of Θ, and the second term f belongs to L 1 (Q).