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We study the existence of solutions of the nonlinear parabolic problem $\partial u/\partial t-div\left[\right|Du-\Theta \left(u\right){|}^{p-2}(Du-\Theta \left(u\right))]+\alpha \left(u\right)=f$ in ]0,T[ × Ω, $\left(\right|Du-\Theta \left(u\right){|}^{p-2}(Du-\Theta \left(u\right)))·\eta +\gamma \left(u\right)=g$ on ]0,T[ × ∂Ω, u(0,·) = u₀ in Ω, with initial data in L¹. We use a time discretization of the continuous problem by the Euler forward scheme.