CONVERGENCE ESTIMATES FOR SOME ABSTRACT LINEAR SECOND ORDER DIFFERENTIAL EQUATIONS WITH TWO SMALL PARAMETERS

Abstract

In a real Hilbert space H we consider the following singularly perturbed Cauchy problem (Equation presented) where u0, u1 ∈ H, f : [0, T ] → H and ϵ, δ are two small parameters. We study the behavior of the solutions uϵδ to the problem (Pϵδ) in two different cases: (i) when ϵ → 0 and δ ≥ δ0 > 0; (ii) when ϵ → 0 and δ → 0. We obtain a priori estimates of the solutions to the perturbed problem, which are uniform with respect to the parameters, and a relationship between the solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior with respect to the parameters in the neighborhood of t = 0. We describe the boundary layer and the boundary layer function in both cases.