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February  2004, 10(1&2): 337-348. doi: 10.3934/dcds.2004.10.337

Longtime behavior of a viscoelastic Timoshenko beam

M. Grasselli 1, , Vittorino Pata 2, and  Giovanni Prouse 2,

1. 

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

Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, Italy, Italy

Received  January 2002 Revised  February 2003 Published  October 2003

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.
Keywords: Timoshenko beam, attractors., absorbing sets, viscoelasticity.
Mathematics Subject Classification: 35B40, 35B41, 37L15, 45K05, 74D0.
Citation: M. Grasselli, Vittorino Pata, Giovanni Prouse. Longtime behavior of a viscoelastic Timoshenko beam. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 337-348. doi: 10.3934/dcds.2004.10.337
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