Longtime behavior of a viscoelastic Timoshenko beam

M. Grasselli; Vittorino Pata; Giovanni Prouse

doi:10.3934/dcds.2004.10.337

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2004, Volume 10, Issue 1&2: 337-348. Doi: 10.3934/dcds.2004.10.337

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Longtime behavior of a viscoelastic Timoshenko beam

1.
Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano
2.
Dipartimento di Matematica "F. Brioschi", Politecnico di Milano

Received: December 31, 2001

Revised: January 31, 2003

Published: December 2003

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Abstract

We consider a Timoshenko model of a viscoelastic beam fixed at the endpoints and subject to nonlinear external forces. The model consists of two coupled second order linear integrodifferential hyperbolic equations that govern the evolution of the lateral displacement $u$ and the total rotation angle $\phi$. We prove that these equations generate a dissipative dynamical system, whose trajectories are eventually confined in a uniform absorbing set, the dissipativity being due to the memory mechanism solely. This fact allows us to state the existence of a uniform compact attractor.

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Mathematics Subject Classification: 35B40, 35B41, 37L15, 45K05, 74D05.

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