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BLOW UP OF SOLUTIONS FOR A TIMOSHENKO EQUATION WITH DAMPING TERMS

Year 2018, , 70 - 80, 27.12.2018
https://doi.org/10.23884/mejs.2018.4.2.03

Abstract

 In this work, we studied the following equation


u_{tt}+△²u-M(‖∇u‖²)△u-△u_{t}+u_{t}=|u|^{q-1}u


 regard to initial and Dirichlet boundary condition. We show that the blow up of solutions with positive and negative initial energy. 

References

  • Adams, R.A., Fournier, J.J.F. (2003). Sobolev Spaces, Academic Press, New York.
  • Chen, W., Zhou, Y. (2009). Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal., 70, 3203-3208.
  • Doshi,C. (1979). On the Analysis of the Timoshenko Beam Theory with and without internal damping, Thesis, Rochester Institute of Technology.
  • Esquivel-Avila, J.A. (2010). Dynamic analysis of a nonlinear Timoshenko equation, Abstract and Applied Analysis, 2011, 1-36.
  • Esquivel-Avila, J.A. (2013). Global attractor for a nonlinear Timoshenko equation with source terms, Mathematical Sciences, 1-8.
  • Georgiev, V., Todorova, G. (1994). Existence of a solution of the wave equation with nonlinear damping and source term, Journal of Differential Equations, 109, 295-308.
  • Levine, H.A. (1974). Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu_{tt}=-Au+F(u), Trans. Amer. Math. Soc., 192, 1-21.
  • Levine, H.A. (1974). Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM Journal on Applied Mathematics, 5, 138-146.
  • Li, M.R., Tsai, L.Y. (2003). Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal., 54 (8), 1397-1415.
  • Messaoudi, S.A. (2001). Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231, 105-111.
  • Messaoudi, S.A. (2002). Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265(2), 296-308.
  • Pişkin, E. (2015). Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math., 13, 408-420.
  • Pişkin E., Irkıl, N. (2016). Blow up of Positive Initial-Energy Solutions for the Extensible Beam Equation with Nonlinear Damping and Source terms, Ser. Math. Inform., 31(3), 645-654.
  • Timoshenko, S.P. (1921). On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars, Philosophical Magazine and Journal of Science, 6(41), 744-746.
  • Vitillaro, E. (1999). Global existence theorems for a class of evolution equations with dissipation, Arch. Rational Mech. Anal., 149, 155-182.
  • Wu, S.T., Tsai, L.Y. (2009). On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math., 13 (2A), 545-558.
Year 2018, , 70 - 80, 27.12.2018
https://doi.org/10.23884/mejs.2018.4.2.03

Abstract

References

  • Adams, R.A., Fournier, J.J.F. (2003). Sobolev Spaces, Academic Press, New York.
  • Chen, W., Zhou, Y. (2009). Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal., 70, 3203-3208.
  • Doshi,C. (1979). On the Analysis of the Timoshenko Beam Theory with and without internal damping, Thesis, Rochester Institute of Technology.
  • Esquivel-Avila, J.A. (2010). Dynamic analysis of a nonlinear Timoshenko equation, Abstract and Applied Analysis, 2011, 1-36.
  • Esquivel-Avila, J.A. (2013). Global attractor for a nonlinear Timoshenko equation with source terms, Mathematical Sciences, 1-8.
  • Georgiev, V., Todorova, G. (1994). Existence of a solution of the wave equation with nonlinear damping and source term, Journal of Differential Equations, 109, 295-308.
  • Levine, H.A. (1974). Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu_{tt}=-Au+F(u), Trans. Amer. Math. Soc., 192, 1-21.
  • Levine, H.A. (1974). Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM Journal on Applied Mathematics, 5, 138-146.
  • Li, M.R., Tsai, L.Y. (2003). Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal., 54 (8), 1397-1415.
  • Messaoudi, S.A. (2001). Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231, 105-111.
  • Messaoudi, S.A. (2002). Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265(2), 296-308.
  • Pişkin, E. (2015). Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math., 13, 408-420.
  • Pişkin E., Irkıl, N. (2016). Blow up of Positive Initial-Energy Solutions for the Extensible Beam Equation with Nonlinear Damping and Source terms, Ser. Math. Inform., 31(3), 645-654.
  • Timoshenko, S.P. (1921). On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars, Philosophical Magazine and Journal of Science, 6(41), 744-746.
  • Vitillaro, E. (1999). Global existence theorems for a class of evolution equations with dissipation, Arch. Rational Mech. Anal., 149, 155-182.
  • Wu, S.T., Tsai, L.Y. (2009). On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math., 13 (2A), 545-558.
There are 16 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Article
Authors

Erhan Pişkin 0000-0001-6587-4479

Hazal Yüksekkaya This is me 0000-0002-1863-2909

Publication Date December 27, 2018
Submission Date November 2, 2018
Acceptance Date December 4, 2018
Published in Issue Year 2018

Cite

IEEE E. Pişkin and H. Yüksekkaya, “BLOW UP OF SOLUTIONS FOR A TIMOSHENKO EQUATION WITH DAMPING TERMS”, MEJS, vol. 4, no. 2, pp. 70–80, 2018, doi: 10.23884/mejs.2018.4.2.03.

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