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Arched beams of Bresse type: observability and application in thermoelasticity

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Abstract

This is the first paper of a trilogy intended by the authors in what concerns a unified approach to the stability of thermoelastic arched beams of Bresse type under Fourier’s law. Our main goal in this starting work is to develop an original observability inequality for conservative Bresse systems with non-constant coefficients. Then, as a powerful application, we prove mathematically that the stability of a partially damped model in thermoelastic Bresse beams is invariant under the boundary conditions. The exponential and optimal polynomial decay rates are addressed. This approach gives a new view on the stability of Bresse systems subject to different boundary conditions as well as it provides an accurate answer for the related issue raised by Liu and Rao (Z. Angew. Math. Phys. 60(1): 54–69, 2009) from both the physical and mathematical points of view.

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Notes

  1. Throughout this paper, the notion of semi-uniform stability is always invoked when the stability of the semigroup solution does not occur for all weak initial data (say at the same energy level of solutions), but only for more regular initial data, e.g., data in the domain of the infinitesimal generator of the semigroup.

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Authors and Affiliations

  1. Department of Mathematics, State University of Maringá, Maringá, Paraná, 87020-900, Brazil

    Gabriel E. Bittencourt Moraes

  2. Department of Mathematics, State University of Londrina, Londrina, Paraná, 86057-970, Brazil

    Marcio A. Jorge Silva

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  1. Gabriel E. Bittencourt Moraes
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  2. Marcio A. Jorge Silva
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Correspondence to Marcio A. Jorge Silva.

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Gabriel E. Bittencourt Moraes: Supported by the CAPES. Finance code 001. (Master and Ph.D. Scholarships) Marcio A. Jorge Silva: Partially supported by the CNPq, Grant 301116/2019-9.

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Moraes, G.E.B., Silva, M.A.J. Arched beams of Bresse type: observability and application in thermoelasticity. Nonlinear Dyn 103, 2365–2390 (2021). https://doi.org/10.1007/s11071-021-06243-3

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