Convergence estimates for abstract second order differential equations with two small parameters and Lipschitzian nonlinearities

Perjan, Andrei and Rusu, Galina

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

    \[\left\{ \begin{array}{l} \varepsilon u^{\prime\prime}_{\varepsilon\delta}(t)+\d\,u^{\prime}_{\varepsilon\delta}(t)+ A u_{\varepsilon\delta}(t)+ B\big(u_{\varepsilon\delta}(t)\big)=f(t), \quad t \in (0, T), \\ u_{\varepsilon\delta}(0)=u_{0},\quad u^{\prime}_{\varepsilon\delta}(0)=u_{1},\ \end{array} \right.\]

where u_0, u_1\in H, f:[0,T]\mapsto H, \varepsilon,
\delta are two small parameters, A is a linear self-adjoint operator and B is a nonlinear A^{1/2} Lipschitzian operator.

We study the behavior of solutions u_{\varepsilon\delta} 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  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 t=0.

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Perjan, Andrei, Rusu, Galina