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Stability of weakly dissipative Reissner–Mindlin–Timoshenko plates: A sharp result

  • A. D. S. CAMPELO (a1), D. S. ALMEIDA JÚNIOR (a1) and M. L. SANTOS (a1)
Abstract

In the present article, we show that there exists a critical number that stabilizes the Reissner–Mindlin–Timoshenko system with frictional dissipation acting on rotation angles. We identify two speed characteristics v 1 2:=K1 and v 2 2:=D2, and we show that the system is exponentially stable if and only if \begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*} For v 1 2v 2 2, we prove that the system is polynomially stable and determine an optimal estimate for the decay. To confirm our analytical results, we compute the numerical solutions by means of several numerical experiments by using a finite difference method.

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Dedicated to Prof. Marcelo Moreira Cavalcanti on the occasion of his 60th Birthday.

The first author thanks to PNPD/CAPES for his financial support. The second and third authors thank to IMPA for its hospitality during their stage visiting professor. The second author is supported by CNPq Grant 311553/2013-3 and by CNPq Grant 458866/2014-8 (Universal Project -2014). The third author is supported by CNPq Grant 302899/2015-4 and by CNPq Grant 401769/2016-0 (Universal Project -2016).

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] D. S. Almeida Júnior & J. E. Muñoz Rivera (2015) Stability criterion to explicit finite difference applied to the Bresse system. Afrika Mathematika 26 (5), 761778.

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[8] A. Campelo , D. Almeida Júnior & M. L. Santos (2016) Stability to the dissipative Reissner-Mindlin-Timoshenko acting on displacement equation. Eur. J. Appl. Math. 27 (2), 157193.

[10] M. Grobbelaar-Van Dalsen (2011) Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound. Appl. Anal. 90 (9), 14191449.

[12] M. Grobbelaar-Van Dalsen (2015) Polynomial decay rate of a thermoelastic Mindlin–Timoshenko plate model with Dirichlet boundary conditions. Z. Angew. Math. Phys. 66 (1), 113128.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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