Global and exponential attractors for the singularly perturbedextensible beam

Michele Coti Zelati

doi:10.3934/dcds.2009.25.1041

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2009, Volume 25, Issue 3: 1041-1060. Doi: 10.3934/dcds.2009.25.1041

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Global and exponential attractors for the singularly perturbed extensible beam

Michele Coti Zelati^1,

1.
Politecnico di Milano - Dipartimento di Matematica "F.Brioschi", Via Bonardi 9, 20133 Milano

Received: October 31, 2008

Revised: February 28, 2009

Published: August 2009

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Abstract

The paper deals with the nonlinear evolution equation

ε∂_ttu + $\delta$∂_tu+ $\omega$∂_xxxx u-$[\beta+\int_0^1[\partial_y u(y,t)]^2\d y ]$∂_xxu=f,

which describes the motion of the vertical deflection of an extensible Kirchhoff beam. The existence of the global attractor of optimal regularity is shown, as well as the existence of a family of exponential attractors Hölder-continuous in the symmetric Hausdorff distance (with respect to ε) and of finite fractal dimension uniformly bounded with respect to ε.

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Mathematics Subject Classification: 35B25, 35B41, 35B65, 37B25, 74K05, 74K10.

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