Limiting behavior of global attractors for singularly perturbed beam equations with strong damping

Daniel Ševčovič

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 1, page 45-60
  • ISSN: 0010-2628

Abstract

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The limiting behavior of global attractors 𝒜 ε for singularly perturbed beam equations ε 2 ∂ 2 u ∂ t 2 + ε δ ∂ u ∂ t + A ∂ u ∂ t + α A u + g ( ∥ u ∥ 1 / 4 2 ) A 1 / 2 u = 0 is investigated. It is shown that for any neighborhood 𝒰 of 𝒜 0 the set 𝒜 ε is included in 𝒰 for ε small.

How to cite

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Ševčovič, Daniel. "Limiting behavior of global attractors for singularly perturbed beam equations with strong damping." Commentationes Mathematicae Universitatis Carolinae 32.1 (1991): 45-60. <http://eudml.org/doc/247247>.

@article{Ševčovič1991,
abstract = {The limiting behavior of global attractors $\mathcal \{A\}_\varepsilon $ for singularly perturbed beam equations \[\varepsilon ^2 \frac\{\partial ^2u\}\{\partial t^2\}+ \varepsilon \delta \frac\{\partial u\}\{\partial t\}+A \frac\{\partial u\}\{\partial t\}+\alpha Au+g(\Vert u\Vert \_\{1/4\}^2)A^\{1/2\}u=0 \] is investigated. It is shown that for any neighborhood $\mathcal \{U\}$ of $\mathcal \{A\}_0$ the set $\mathcal \{A\}_\varepsilon $ is included in $\mathcal \{U\}$ for $\varepsilon $ small.},
author = {Ševčovič, Daniel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {strongly damped beam equation; compact attractor; upper semicontinuity of global attractors; strongly damped beam equation; compact attractor; upper semicontinuity of global attractors},
language = {eng},
number = {1},
pages = {45-60},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Limiting behavior of global attractors for singularly perturbed beam equations with strong damping},
url = {http://eudml.org/doc/247247},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Ševčovič, Daniel
TI - Limiting behavior of global attractors for singularly perturbed beam equations with strong damping
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 1
SP - 45
EP - 60
AB - The limiting behavior of global attractors $\mathcal {A}_\varepsilon $ for singularly perturbed beam equations \[\varepsilon ^2 \frac{\partial ^2u}{\partial t^2}+ \varepsilon \delta \frac{\partial u}{\partial t}+A \frac{\partial u}{\partial t}+\alpha Au+g(\Vert u\Vert _{1/4}^2)A^{1/2}u=0 \] is investigated. It is shown that for any neighborhood $\mathcal {U}$ of $\mathcal {A}_0$ the set $\mathcal {A}_\varepsilon $ is included in $\mathcal {U}$ for $\varepsilon $ small.
LA - eng
KW - strongly damped beam equation; compact attractor; upper semicontinuity of global attractors; strongly damped beam equation; compact attractor; upper semicontinuity of global attractors
UR - http://eudml.org/doc/247247
ER -

References

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  7. Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer Verlag. Zbl0663.35001MR0610244
  8. Hale J.K., Rougel G., Upper semicontinuity of an attractor for a singularly perturbed hyperbolic equations, J. of Diff. Equations 73 (1988), 197-215. (1988) MR0943939
  9. 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. (1990) MR1041197
  10. Massat P., Limiting behavior for strongly damped nonlinear wave equations, J. of Diff. Equations 48 (1983), 334-349. (1983) MR0702424
  11. Massat P., Attractivity properties of α -contractions, J. of Diff. Equations 48 (1983), 326-333. (1983) MR0702423

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