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Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation. (English) Zbl 0666.35012

For a smooth bounded domain \(\Omega \subset R^ n\), \(n\leq 3\), and \(\epsilon\) a real parameter, consider the hyperbolic equation \[ \epsilon u_{tt}+u_ t-\Delta u=-f(u)-g\quad in\quad \Omega \] with Dirichlet boundary conditions. Under certain conditions on the function f(u), this equation has a compact attractor \({\mathcal A}_{\epsilon}\) in \(H^ 1_ 0\times L^ 2\). For \(\epsilon =0\), the parabolic equation also has a compact attractor which can be naturally embedded into a compact set \({\mathcal A}_ 0\) in \(H^ 1_ 0\times L_ 2\). The authors demonstrate that, for any neighborhood of U, the set \({\mathcal A}_{\epsilon}\subset U\) for \(\epsilon\) small.
Reviewer: M.Witten

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B25 Singular perturbations in context of PDEs
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35L70 Second-order nonlinear hyperbolic equations
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