In a real Hilbert space H consider the following singularly perturbed Cauchy problem (Formula presented), where A(t): V ⊂ H → H, t ∈ [0, ∞), is a family of linear self-adjoint operators, u₀, u₁ ∈ H, f: [0, T ] ↦→ H and ε, δ are two small parameters. We study the behavior of solutions u_εδ to this problem in two different cases: ε → 0 and δ ≥ δ₀ > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem. 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 perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0. We show the boundary layer and boundary layer function in both cases.