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

Perjan, Andrei and Rusu, Galina



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.

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Perjan, Andrei, Rusu, Galina