Abstract: In a real Hilbert space H we consider the following singularly perturbed Cauchy problem
(Pεδ)εuεδ″(t)+δuεδ′(t)+Auεδ(t)=f(t),t∈(0,T),uεδ(0)=u0,uεδ′(0)=u1,
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.
Keywords: singular perturbation, abstract second order Cauchy problem, boundary layer function, a priori estimate
