Abstract
This paper is concerned with the Timoshenko system, a recognized model for vibrations of thin prismatic beams. The corresponding autonomous system has been widely studied. However, there are only a few works dedicated to its non-autonomous counterpart. Here, we investigate the long-time dynamics of Timoshenko systems involving a nonlinear foundation and subjected to perturbations of time-dependent external forces. The main result establishes the existence of a pullback exponential attractor, which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. The upper-semicontinuity of attractors, as the non-autonomous forces tend to zero, is also studied.
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Acknowledgements
This work was partially supported by CNPq (Brazil) Grant 310041/2015-5. It was initiated while the third author was a long-term visitor at ICMC-USP, from August 2015 to July 2016. The authors thank Professor Alexandre N. Carvalho for some interesting conversation on the subject. They also thank the referees for their useful comments and suggestions.
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Ma, T.F., Monteiro, R.N. & Pereira, A.C. Pullback Dynamics of Non-autonomous Timoshenko Systems. Appl Math Optim 80, 391–413 (2019). https://doi.org/10.1007/s00245-017-9469-2
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DOI: https://doi.org/10.1007/s00245-017-9469-2