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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose


Mathematical Citation Quotient (MCQ) 2017: 0.71

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2353-0626
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Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement

Aissa Guesmia
  • Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, 3 Rue Augustin Fresnel, BP 45112, 57073 Metz Cedex 03, France
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Published Online: 2017-11-07 | DOI: https://doi.org/10.1515/msds-2017-0008

Abstract

The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was obtained by Guesmia and Kafini [10] (three infinite memories), Guesmia and Kirane [11] (two infinite memories), Guesmia [9] (one infinite memory acting on the longitudinal displacement) and De Lima Santos et al. [6] (one infinite memory acting on the shear angle displacement). When the kernel functions have an exponential decay at infinity, the obtained stability estimates in these papers lead to the exponential stability of the system if the speeds ofwave propagations are the same, and to the polynomial one with decay rate otherwise. The subject of this paper is to study the case where only one infinite memory is considered and it is acting on the vertical displacement. As far as we know, this case has never studied before in the literature. We show that this case is deeply different from the previous ones cited above by proving that the exponential stability does not hold even if the speeds of wave propagations are the same and the kernel function has an exponential decay at infinity. Moreover, we prove that the system is still stable at least polynomially where the decay rate depends on the smoothness of the initial data. For classical solutions, this decay rate is arbitrarily close to . The proof is based on a combination of the energy method and the frequency domain approach to overcome the new mathematical difficulties generated by our system.

Keywords: Bresse system; Infinite memory; Asymptotic behavior; Energy method; Frequency domain approach

MSC 2010: 35B40; 35L45; 74H40; 93D20; 93D15

References

  • [1] M. Afilal, A. Guesmia and A. Soufyane, On the exponential and polynomial stability for a linear thermoelastic Bresse system with second sound, submitted.Google Scholar

  • [2] F. Alabau-Boussouira, J. E.Muñoz Rivera and D. S. Almeida Júnior, Stability to weak dissipative Bresse system, J.Math. Anal. Appl., 374 (2011), 481-498.Google Scholar

  • [3] M. O. Alves, L. H. Fatori, M. A. Jorge Silva and R. N. Monteiro, Stability and optimality of decay rate for weakly dissipative Bresse system, Math. Meth. Appl. Sci., 38 (2015), 898-908.Web of ScienceGoogle Scholar

  • [4] J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, Paris, 1859.Google Scholar

  • [5] W. Charles, J. A. Soriano, F. A. Nascimento and J. H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, J. Diff. Equa., 255 (2013), 2267-2290.Web of ScienceGoogle Scholar

  • [6] M. De Lima Santos, A. Soufyane and D. Da Silva Almeida Júnior, Asymptotic behavior to Bresse system with past history, Quart. Appl. Math., 73 (2015), 23-54.Google Scholar

  • [7] L. H. Fatori and R. N. Monteiro, The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25 (2012), 600-604.CrossrefWeb of ScienceGoogle Scholar

  • [8] L. H. Fatori and J. M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904.Web of ScienceGoogle Scholar

  • [9] A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Medi. J. Math., 14 (2017), 19 pages.Google Scholar

  • [10] A. Guesmia and M. Kafini, Bresse system with infinite memories, Math. Meth. Appl. Sci., 38 (2015), 2389-2402.Web of ScienceGoogle Scholar

  • [11] A. Guesmia and M. Kirane, Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67 (2016), 1-39.Web of ScienceGoogle Scholar

  • [12] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Equa., 1 (1985), 43-56.Google Scholar

  • [13] A. Keddi, T. Apalara and S. Messaoudi, Exponential and polynomial decay in a thermoelastic-Bresse system with second sound, Appl. Math. Optim, DOI: 10.1007/s00245-016-9376-y, to appear. CrossrefGoogle Scholar

  • [14] J. E. Lagnese, G. Leugering and J. P. Schmidt, Modelling of dynamic networks of thin thermoelastic beams,Math. Meth. Appl. Scie., 16 (1993), 327-358.Google Scholar

  • [15] J. E. Lagnese, G. Leugering and J. P. Schmidt, Modelling Analysis and Control of Dynamic Elastic Multi-Link Structures, Systems Control Found. Appl., 1994.Google Scholar

  • [16] Z. Liu and B. Rao, Characterization of polymomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.CrossrefGoogle Scholar

  • [17] Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60 (2009), 54-69.CrossrefGoogle Scholar

  • [18] N. Najdi and A. Wehbe, Weakly locally thermal stabilization of Bresse systems, Elec. J. Diff. Equa., 2014 (2014), 1-19.Google Scholar

  • [19] N. Noun and A. Wehbe, Weakly locally internal stabilization of elastic Bresse system, C. R. Acad. Scie. Paris, Sér. I, 350 (2012), 493-498.Google Scholar

  • [20] J. Pruss, On the spectrum of C0 semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.Google Scholar

  • [21] J. A. Soriano, W. Charles and R. Schulz, Asymptotic stability for Bresse systems, J. Math. Anal. Appl., 412 (2014), 369-380.Google Scholar

  • [22] J. A. Soriano, J. M. Rivera and L. H. Fatori, Bresse system with indefinite damping, J. Math. Anal. Appl., 387 (2012), 284-290.Google Scholar

  • [23] A. Soufyane and B. Said-Houari, The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equa. Cont. Theory, 3 (2014), 713-738.Google Scholar

  • [24] A. Wehbe and W. Youssef, Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51 (2010), 1-17.Web of ScienceGoogle Scholar

About the article

Received: 2017-09-14

Accepted: 2017-10-07

Published Online: 2017-11-07

Published in Print: 2017-10-26


Citation Information: Nonautonomous Dynamical Systems, Volume 4, Issue 1, Pages 78–97, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2017-0008.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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