In a real Hilbert space H we consider the following perturbed Cauchy problem [formula presented], where u₀, u₁ ∈ H, f: [0, T] ↦ H and ε, δ are two small parameters, A is a linear self-adjoint operator, B is a locally Lipschitz and monotone operator. We study the behavior of solutions u_εδ to the problem (P_εδ) in two different cases: (i) when ε → 0 and δ ≥ δ₀ > 0; (ii) when ε → 0 and δ → 0. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighborhood of t = 0. We show the boundary layer and boundary layer function in both cases.