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 $$\begin{array}{}\text{(}{\mathit{P}}_{\mathit{\epsilon }\mathit{\delta }}\text{)}& \left\{\begin{array}{cc}\mathit{\epsilon }{\mathit{u}}_{\mathit{\epsilon }\mathit{\delta }}^{″}\mathrm{(}\mathit{t}\mathrm{)}+\mathit{\delta }{\mathit{u}}_{\mathit{\epsilon }\mathit{\delta }}^{\prime }\mathrm{(}\mathit{t}\mathrm{)}+\mathit{A}{\mathit{u}}_{\mathit{\epsilon }\mathit{\delta }}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{f}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\hfill & \mathit{t}\in \mathrm{(}0\mathrm{,}\mathit{T}\mathrm{)}\mathrm{,}\hfill \\ {\mathit{u}}_{\mathit{\epsilon }\mathit{\delta }}\mathrm{(}0\mathrm{)}={\mathit{u}}_0\mathrm{,}{\mathit{u}}_{\mathit{\epsilon }\mathit{\delta }}^{\prime }\mathrm{(}0\mathrm{)}={\mathit{u}}_1\mathrm{,}\hfill \end{array}\right.\end{array}$$ where ${\mathit{u}}_0\mathrm{,}{\mathit{u}}_1\in \mathit{H}$ , $\mathit{f}:[0\mathrm{,}\mathit{T}]\mapsto \mathit{H}$ and ε, δ are two small parameters.

We study the behavior of the solutions ${\mathit{u}}_{\mathit{\epsilon }\mathit{\delta }}$ to the problem ( P ε δ ) in two different cases:

when $\mathit{\epsilon }\to 0$ and $\mathit{\delta }⩾{\mathit{\delta }}_0\mathrm{>}0$ ;

when $\mathit{\epsilon }\to 0$ and $\mathit{\delta }\to 0$ .

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 $\mathit{t}=0$ . We describe the boundary layer and the boundary layer function in both cases.