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Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms

Received: 21 August 2016     Accepted: 29 August 2016     Published: 14 October 2016
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Abstract

In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 2, Issue 4)
DOI 10.11648/j.ijamtp.20160204.13
Page(s) 41-45
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Timoshenko System, Source Term, Damping Term, Blow-up

References
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[3] S. A. Messaoudi, B. Said-Houari. Uniform decay in a Timoshenko system with past history. J. Math. Anal. Appl. 360 (2009), 458-475.
[4] N. E. Tatar. Stabilization of a viscoelastic Timoshenko beam. Applicable Analysis, 92 (1) (2013), 27-43.
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[6] S. A. Messaoudi, A. Soufyame. Boundary stabilization of a nonlinear system of Timoshenko type. Nonlinear Analysis, 67 (2007), 2107-2112.
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[8] F. D. Araruua, J. E. S. Borges. Existence and boundary stabilization of the semilinear Mindlin-Timoshenko system. Electromic J. of Qualitative Theory of Diff. Equa. 34 (2008), 1-27.
[9] I. Chueshov, I. Lasiecka. Global attractors for Mindlin-Tomoshenko plate and for their Kirchhoff limits. Milan J. Math. 74 (2006), 117-138.
[10] C. Gorgi and F. M.Vegni. Uniform energy estimates for a semilinear evolution equation of the Mindlin-Timoshenko beam with memory. Mathematical and Computer Modelling, 39 (2004), 1005-1021.
[11] A. Soufyame, M. Afilal, T. Aouam. General decay of solutions of a nonlinear Timoshenko system with a boundary control of memory type. Differential and Integral Equations, 22 (2009), 1125-1139.
[12] P. Pei, M. A. Rammaha, D. Toundykov. Local and global well-posedness of semilinear Reissner-Mindlin-Timoshenko plate equations. Nonlinear Analysis, 105 (2014), 62-85.
[13] P. Pei, M. A. Rammaha, D. Toundykov. Global well-posedness and stability of semilinear Mindlin-Timoshenko systems. J. Math. Anal. Appl. 418 (2014), 535-568.
[14] D. H. Sattinger. On global solutions of nonlinear hyperbolic equation. Arch. Rational Mech. Anal. 30 (1968), 108-172.
[15] I. E. Payne, D. H. Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22 (1975), 273-303.
[16] K. Agre, M. A. Rammaha. System of nonlinear wave equation with damping and source terms. Differential and Integral Equation, 19 (1) (2006), 1235-1270.
[17] B. Said-Houari. Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms. Differential and Integral Equation, 23 (1-2) (2010), 29-92.
[18] E. Vitillaro. Global nonexistence theorems for a class of evolution equation with dissipations. Arch. Rational Mech Anal. 149 (1999), 155-182.
[19] C. O. Alves, M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Rammaha, D. Toundykov. On existence, uniform decay rates and blow up for solutions of system-s of nonlinear wave equations with damping and source terms. Discrete Contin. Dyn. Syst. Ser. S, 2 (3) (2009), 583-608.
[20] M. A. Rammaha, S. Sakuntasathien. Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms. Nonlinear Analysis: Theory, Methods and Applications, 72 (5) (2005), 2658-2683.
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Cite This Article
  • APA Style

    Jian Dang, Qingying Hu, Hongwei Zhang. (2016). Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. International Journal of Applied Mathematics and Theoretical Physics, 2(4), 41-45. https://doi.org/10.11648/j.ijamtp.20160204.13

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    ACS Style

    Jian Dang; Qingying Hu; Hongwei Zhang. Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. Int. J. Appl. Math. Theor. Phys. 2016, 2(4), 41-45. doi: 10.11648/j.ijamtp.20160204.13

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    AMA Style

    Jian Dang, Qingying Hu, Hongwei Zhang. Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms. Int J Appl Math Theor Phys. 2016;2(4):41-45. doi: 10.11648/j.ijamtp.20160204.13

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  • @article{10.11648/j.ijamtp.20160204.13,
      author = {Jian Dang and Qingying Hu and Hongwei Zhang},
      title = {Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms},
      journal = {International Journal of Applied Mathematics and Theoretical Physics},
      volume = {2},
      number = {4},
      pages = {41-45},
      doi = {10.11648/j.ijamtp.20160204.13},
      url = {https://doi.org/10.11648/j.ijamtp.20160204.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20160204.13},
      abstract = {In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.},
     year = {2016}
    }
    

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    T1  - Blow-up of Solutions for Semilinear Timoshenko System with Damping and Source Terms
    AU  - Jian Dang
    AU  - Qingying Hu
    AU  - Hongwei Zhang
    Y1  - 2016/10/14
    PY  - 2016
    N1  - https://doi.org/10.11648/j.ijamtp.20160204.13
    DO  - 10.11648/j.ijamtp.20160204.13
    T2  - International Journal of Applied Mathematics and Theoretical Physics
    JF  - International Journal of Applied Mathematics and Theoretical Physics
    JO  - International Journal of Applied Mathematics and Theoretical Physics
    SP  - 41
    EP  - 45
    PB  - Science Publishing Group
    SN  - 2575-5927
    UR  - https://doi.org/10.11648/j.ijamtp.20160204.13
    AB  - In this paper, we are concerned with one-dimensional Timoshenko model for a beam with nonlinear damping and source terms. We establish a blow-up result when the initial energy is positive and the initial data is not in a potential well. Under arbitrary positive initial energy, we also prove a finite-time blow-up result for a special case.
    VL  - 2
    IS  - 4
    ER  - 

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Author Information
  • Department of Mathematics, Henan University of Technology, Zhengzhou, China

  • Department of Mathematics, Henan University of Technology, Zhengzhou, China

  • Department of Mathematics, Henan University of Technology, Zhengzhou, China

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