Global and exponential attractors for the singularly perturbed extensible beam

Pages: 1041 - 1060, Volume 25, Issue 3, November 2009

doi:10.3934/dcds.2009.25.1041       Abstract        Full Text (249.7K)

Michele Coti Zelati - Politecnico di Milano - Dipartimento di Matematica "F.Brioschi", Via Bonardi 9, 20133 Milano, Italy (email)

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.

Keywords:  Extensible beam, global attractor, exponential attractors.
Mathematics Subject Classification:  35B25, 35B41, 35B65, 37B25, 74K05, 74K10.

Received: November 2008;      Revised: March 2009;      Published: August 2009.