Geodesic
Limiting behavior of global attractors for singularly perturbed beam equations with strong damping
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 45-60 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The limiting behavior of global attractors $\Cal A_\varepsilon $ for singularly perturbed beam equations $$\varepsilon^2 \frac{\partial^2u}{\partial t^2}+ \varepsilon\delta \frac{\partial u}{\partial t}+A \frac{\partial u}{\partial t}+\alpha Au+g(\|u\|_{1/4}^2)A^{1/2}u=0 $$ is investigated. It is shown that for any neighborhood $\Cal U$ of $\Cal A_0$ the set $\Cal A_\varepsilon$ is included in $\Cal U$ for $\varepsilon$ small.
The limiting behavior of global attractors $\Cal A_\varepsilon $ for singularly perturbed beam equations $$\varepsilon^2 \frac{\partial^2u}{\partial t^2}+ \varepsilon\delta \frac{\partial u}{\partial t}+A \frac{\partial u}{\partial t}+\alpha Au+g(\|u\|_{1/4}^2)A^{1/2}u=0 $$ is investigated. It is shown that for any neighborhood $\Cal U$ of $\Cal A_0$ the set $\Cal A_\varepsilon$ is included in $\Cal U$ for $\varepsilon$ small.
Classification : 35B25, 35B40, 35Q20, 35Q72, 37C70, 47H20, 73K05, 74H45, 74K10
Keywords: strongly damped beam equation; compact attractor; upper semicontinuity of global attractors 
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     author = {\v{S}ev\v{c}ovi\v{c}, Daniel},
     title = {Limiting behavior of global attractors  for singularly perturbed beam equations  with strong damping},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {45--60},
     year = {1991},
     volume = {32},
     number = {1},
     mrnumber = {1118289},
     zbl = {0741.35089},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1991_32_1_a6/}
}
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Ševčovič, Daniel. Limiting behavior of global attractors  for singularly perturbed beam equations  with strong damping. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 45-60. http://geodesic.mathdoc.fr/item/CMUC_1991_32_1_a6/

[BV] Babin A.B., Vishik M.N.: Attraktory evolucionnych uravnenij s častnymi proizvodnymi i ocenki ich razmernosti (in Russian). Uspechi mat. nauk 38 (1983), 133-185. | MR

[B1] Ball J.M.: Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 42 (1973), 61-96. | MR | Zbl

[B2] Ball J.M.: Stability theory for an extensible beam. J. of Diff. Equations 14 (1973), 399-418. | MR | Zbl

[ChL] Chow S.-N., Lu K.: Invariant manifolds for flows in Banach spaces. J. of Diff. Equations 74 (1988), 285-317. | MR | Zbl

[F] Fitzgibbon W.E.: Strongly damped quasilinear evolution equations. J. of Math. Anal. Appl. 79 (1981), 536-550. | MR | Zbl

[GT] Ghidaglia J.M., Temam R.: Attractors for damped nonlinear hyperbolic equations. J. de Math. Pures et Appl. 79 (1987), 273-319. | MR | Zbl

[H] Henry D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840, Springer Verlag. | MR | Zbl

[HR1] Hale J.K., Rougel G.: Upper semicontinuity of an attractor for a singularly perturbed hyperbolic equations. J. of Diff. Equations 73 (1988), 197-215. | MR

[HR2] Hale J.K., Rougel G.: Lower semicontinuity of an attractor for a singularly perturbed hyperbolic equations. Journal of Dynamics and Diff. Equations 2 (1990), 16-69. | MR

[M1] Massat P.: Limiting behavior for strongly damped nonlinear wave equations. J. of Diff. Equations 48 (1983), 334-349. | MR

[M2] Massat P.: Attractivity properties of $\alpha $-contractions. J. of Diff. Equations 48 (1983), 326-333. | MR