September  2009, 25(3): 1041-1060. doi: 10.3934/dcds.2009.25.1041

Global and exponential attractors for the singularly perturbed extensible beam

1. 

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

Received  November 2008 Revised  March 2009 Published  August 2009

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.

Citation: Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041
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