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


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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[1] Babin, A.V.; Vishik, M.I., Regular attractors of semigroups and evolution equations, J. math. pures appl., 62, 441-491, (1983) · Zbl 0565.47045
[2] Babin, A.V.; Vishik, M.I., Attracteurs maximaux dans LES équations aux dérivées partielles (notes rédigées par A. haraux), (1984), Collège de France, Pittman, 1985
[3] Billotti, J.E.; LaSalle, J.P., Periodic dissipative processes, Bull. amer. math. soc., 6, 1082-1089, (1971) · Zbl 0274.34061
[4] Ghidaglia, J.M.; Témam, R., Attractors for damped nonlinear hyperbolic equations, J. math. pures appl., 66, 273-319, (1987) · Zbl 0572.35071
[5] Hale, J.K., Asymptotic behavior and dynamics in infinite dimensions, (), 1-42 · Zbl 0653.35006
[6] Haraux, A., Two remarks on dissipative hyperbolic problems, () · Zbl 0579.35057
[7] Henry, D., Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[8] Ladyzhenskaya, O.A., The boundary value problems of mathematical physics, () · Zbl 0164.12501
[9] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod, Gauthier-Villars Paris · Zbl 0189.40603
[10] Lions, J.L.; Magenes, E., Problèmes aux limites non homogènes et applications, (1968), Dunod Paris · Zbl 0165.10801
[11] {\scO. Lopes, and S. Ceron}, Existence of forced periodic solutions of dissipative semilinear hyperbolic equations and systems, Preprint of the University of Campinas (UNICAMP), São Paulo, Brazil.
[12] Massatt, P., Attractivity properties of α-contractions, J. differential equations, 48, 326-333, (1983) · Zbl 0542.34058
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