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Regular approach to the problem of attractors of singularly perturbed equations. (Russian. English summary) Zbl 0729.34034

Author’s summary: “Gradient semidynamical systems, which depend on parameter(s) \(\lambda\) and possess a finite number of hyperbolic equilibrium points, are considered. Under certain assumptions it is proved that the global attractor \({\mathcal M}_{\lambda}\) is Hölder continuous in \(\lambda\) in the Hausdorff metric. As an intermediate result it is shown that \({\mathcal M}_{\lambda}\) uniformly in \(\lambda\) exponentially attracts every bounded set. The results are applied to prove the convergence (in the Hausdorff metric) of the global attractor of an abstract damped hyperbolic equation with a small parameter \(\epsilon\) at the second-order time derivative - to the attractor of a corresponding parabolic equation”.

MSC:

34D45 Attractors of solutions to ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34E15 Singular perturbations for ordinary differential equations
35B25 Singular perturbations in context of PDEs
35L99 Hyperbolic equations and hyperbolic systems
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