carpathian_2022_38_1_179_200In a real Hilbert space 
we consider the following singularly perturbed Cauchy problem
![\[\left\{ \begin{array}{l} \varepsilon u^{\prime\prime}_{\varepsilon\delta}(t)+\d\,u^{\prime}_{\varepsilon\delta}(t)+ A u_{\varepsilon\delta}(t)+ B\big(u_{\varepsilon\delta}(t)\big)=f(t), \quad t \in (0, T), \\ u_{\varepsilon\delta}(0)=u_{0},\quad u^{\prime}_{\varepsilon\delta}(0)=u_{1},\ \end{array} \right.\]](https://www.carpathian.cunbm.utcluj.ro/wp-content/ql-cache/quicklatex.com-903c2632959dcc32108c8cf4dfcc0094_l3.png "Rendered by QuickLaTeX.com")
where 
, ![f:[0,T]\mapsto H,](https://www.carpathian.cunbm.utcluj.ro/wp-content/ql-cache/quicklatex.com-c2764db817246280f9ee38ce0846fb96_l3.png "Rendered by QuickLaTeX.com")
are two small parameters, 
is a linear self-adjoint operator and 
is a nonlinear 
Lipschitzian operator.
We study the behavior of solutions 
in two different cases:
and 
and 
relative to solution to the corresponding unperturbed problem.
We obtain some 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 unperturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of 
