BLOW UP OF SOLUTIONS FOR A TIMOSHENKO EQUATION WITH DAMPING TERMS

Year 2018, Volume: 4 Issue: 2, 70 - 80, 27.12.2018

Erhan Pişkin Hazal Yüksekkaya

https://doi.org/10.23884/mejs.2018.4.2.03

Cited By: 3

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. 

Keywords

Timoshenko equation, Blow up, Damping term

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.