Andrei Perjan
Moldova State University, Chişinău, Moldova

In the real Hilbert space $H$ we consider the following Cauchy problem: $$ \left\{ \begin{array}{l} \varepsilon u''_{\varepsilon\delta}(t)+\delta\,u'_{\varepsilon\delta}(t)+ A u_{\varepsilon\delta}(t)+ B\big(u_{\varepsilon\delta}(t)\big)=f_{\varepsilon}(t), \quad t \in (0, T), \\ u_{\varepsilon\delta}(0)=u_{0\varepsilon},\quad u'_{\varepsilon\delta}(0)=u_{1\varepsilon},\ \end{array} \right.\ (Eq\, P_{\varepsilon \delta}) $$ where $A:V\subset H\to H$, be a linear self-adjoint operator and $B$ is nonlinear $A^{1/2}$ lipschitzian or monotone operator, $u_{0\varepsilon}, u_{1\varepsilon}\in H, f_{\varepsilon}: [0,T] \to H$ and $\varepsilon, \delta$ are two small parameters. We investigate the behavior of solutions $u_{\varepsilon\delta}$ to the problem ($P_{\varepsilon\delta}$) in two different cases:

(i) $\varepsilon\to 0$ and $\delta \geq \delta_0>0 $, relative to the solutions to the following unperturbed system: $$ \left\{ \begin{array}{l} \delta l_\delta'(t)+ A l_\delta(t)+B\big(l_\delta(t)\big)=f(t),\quad t\in(0,T),\\ l_\delta(0)=u_0; \end{array} \right.\ (Eq\, P_{\delta}) $$
(ii) $\varepsilon\to 0$ and $\delta \to 0$, relative to the solutions to the following system: $$ A v(t)+B\big(v(t)\big)= f(t),\quad t\in[0,T)\ (Eq\, P_{0}) $$ The mathematical model ${(P_{\varepsilon \delta})}$ governs various physical processes, which are described by the Klein-Gordon equation, the Sine-Gordon equation, the Plate equation and others equations.