» Abstract linear second order differential equations with two small parameters and depending on time operators

In a real Hilbert space $H$ consider the following singularly perturbed Cauchy problem

$\left\{\begin{array}{l} \varepsilon\,u''_{\varepsilon\delta}(t)+ \delta\,u'_{\varepsilon\delta}(t)+A(t)u_{\varepsilon\delta}(t)= f(t),\quad t\in(0,T),\\ u_{\varepsilon\delta}(0)=u_0,\quad u'_{\varepsilon\delta}(0)=u_1, \end{array} \right.$

where $A(t):V\subset H\to H,$ $t\in [0,\infty),$ is a family of linear self-adjoint
operators, $u_0, u_1\in H$ , $f:[0,T]\mapsto H$ and $\varepsilon,$
$\delta$ are two small parameters. We study the behavior of solutions $u_{\varepsilon\delta}$ to this problem
in two different cases: $\varepsilon\to 0$ and $\delta \geq \delta_0>0;$ $\varepsilon\to 0$ and $\delta \to 0$ , relative to solution to the corresponding unperturbed problem.We obtain some {\it 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.