In a real Hilbert space H we consider the following perturbed Cauchy problem ( " u′′ "(t) +  u′ "(t) + Au "(t) + B(u "(t)) = f(t), t ∈ (0, T ), u "(0) = u0, u′ "(0) = u1, (P ") where u0, u1 ∈ H, f : [0, T ] 7→ H and ",  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 " to the problem (P ") in two different cases: (i) when " → 0 and  ≥ 0 > 0; (ii) when " → 0 and  → 0. 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.