Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic behavior to Bresse system with past history

Authors: Mauro de Lima Santos, A. Soufyane and Dilberto da Silva Almeida Júnior
Journal: Quart. Appl. Math. 73 (2015), 23-54
MSC (2010): Primary 74H40
DOI: https://doi.org/10.1090/S0033-569X-2014-01382-4
Published electronically: October 17, 2014
MathSciNet review: 3322725
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the Bresse system with past history acting in the shear angle displacement. We show the exponential decay of the solution if and only if the wave speeds are the same. On the contrary, we show that the Bresse system is polynomial stable with optimal decay rate. The systems of equations considered here introduce new mathematical difficulties in order to determine the asymptotic behavior. As far as the authors know, there have been no contributions made in this sense.

References [Enhancements On Off] (What's this?)

  • [1] Alexander Borichev and Yuri Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010), no. 2, 455-478. MR 2606945 (2011c:47091), https://doi.org/10.1007/s00208-009-0439-0
  • [2] Haïm Brezis, Analyse fonctionnelle, (French). Théorie et applications. [Theory and applications], Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983. MR 697382 (85a:46001)
  • [3] Constantine M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297-308. MR 0281400 (43 #7117)
  • [4] Mauro Fabrizio and Angelo Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021 (93a:73034)
  • [5] Fa Lun Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations 1 (1985), no. 1, 43-56. MR 834231 (87e:34106)
  • [6] Karl F. Graff, Wave motion in elastic solids, Dover Publications, New York, 1991.
  • [7] J. E. Lagnese, Günter Leugering, and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1994. MR 1279380 (95d:93003)
  • [8] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • [9] Jan Prüss, On the spectrum of $ C_{0}$-semigroups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 847-857. MR 743749 (85f:47044), https://doi.org/10.2307/1999112
  • [10] Fatiha Alabau Boussouira, Jaime E. Muñoz Rivera, and Dilberto da S. Almeida Júnior, Stability to weak dissipative Bresse system, J. Math. Anal. Appl. 374 (2011), no. 2, 481-498. MR 2729236 (2011k:35138), https://doi.org/10.1016/j.jmaa.2010.07.046
  • [11] Larry Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385-394. MR 0461206 (57 #1191)
  • [12] Luci Harue Fatori and Rodrigo Nunes Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett. 25 (2012), no. 3, 600-604. MR 2856041 (2012i:35216), https://doi.org/10.1016/j.aml.2011.09.067
  • [13] Luci Harue Fatori and Jaime E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math. 75 (2010), no. 6, 881-904. MR 2740037 (2011m:35214), https://doi.org/10.1093/imamat/hxq038
  • [14] Juan A. Soriano, Jaime E. Muñoz Rivera, and Luci Harue Fatori, Bresse system with indefinite damping, J. Math. Anal. Appl. 387 (2012), no. 1, 284-290. MR 2845750 (2012j:35233), https://doi.org/10.1016/j.jmaa.2011.08.072
  • [15] Zhuangyi Liu and Bopeng Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys. 60 (2009), no. 1, 54-69. MR 2469727 (2009m:74035), https://doi.org/10.1007/s00033-008-6122-6
  • [16] Jaime E. Muñoz Rivera and Hugo D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (2008), no. 1, 482-502. MR 2370668 (2009b:74062), https://doi.org/10.1016/j.jmaa.2007.07.012
  • [17] Jaime E. Muñoz Rivera and Maria Grazia Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, J. Math. Anal. Appl. 326 (2007), no. 1, 691-707. MR 2277813 (2007g:35254), https://doi.org/10.1016/j.jmaa.2006.03.022
  • [18] M. L. Santos, D. S. Almeida Júnior, and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differential Equations 253 (2012), no. 9, 2715-2733. MR 2959386, https://doi.org/10.1016/j.jde.2012.07.012
  • [19] Nahla Noun and Ali Wehbe, Stabilisation faible interne locale de système élastique de Bresse, C. R. Math. Acad. Sci. Paris 350 (2012), no. 9-10, 493-498 (English, with English and French summaries). MR 2929055, https://doi.org/10.1016/j.crma.2012.04.003
  • [20] Zhuangyi Liu and Songmu Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1681343 (2000c:47080)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 74H40

Retrieve articles in all journals with MSC (2010): 74H40

Additional Information

Mauro de Lima Santos
Affiliation: Faculdade de Matemática-Programa de Pós-Graduação em Matemática e Estatística Universidade Federal do Pará, Campus Universitario do Guamá, Rua Augusto Corrêa 01, Cep 66075-110, Pará, Brazil
Email: ls@ufpa.br

A. Soufyane
Affiliation: Faculty of Engineering and Applied Sciences, ALHOSN University, P.O. Box 38772, Abu Dhabi, United Arab Emirates
Email: asoufyane@hotmail.com

Dilberto da Silva Almeida Júnior
Affiliation: Faculdade de Matemática-Programa de Pós-Graduação em Matemática e Estatística Universidade Federal do Pará, Campus Universitario do Guamá, Rua Augusto Corrêa 01, Cep 66075-110, Pará, Brazil
Email: dilberto@ufpa.br

DOI: https://doi.org/10.1090/S0033-569X-2014-01382-4
Keywords: Bresse system, optimal result
Received by editor(s): December 14, 2012
Published electronically: October 17, 2014
Article copyright: © Copyright 2014 Brown University

American Mathematical Society