Convergence estimates for some abstract linear second order differential equations with two small parameters

Article type: Research Article

Authors: Perjan, Andrei | Rusu, Galina^{; *}

Affiliations: Department of Fundamental Mathematics, Moldova State University, A. Mateevici str. 60, MD 2009, Chisinau, Moldova

Correspondence: [*] Corresponding author. E-mail: rusugalinamoldova@gmail.com.

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

DOI: 10.3233/ASY-161357

Journal: Asymptotic Analysis, vol. 97, no. 3-4, pp. 337-349, 2016