Taiwanese Journal of Mathematics

Blow up Solutions to a System of Higher-order Kirchhoff-type Equations with Positive Initial Energy

Amir Peyravi

Full-text: Open access

Abstract

In this paper we investigate blow up property of solutions for a system of nonlinear higher order Kirchhoff equations with nonlinear dissipations and positive initial energy. Some estimates for lower bound of the blow up time are also given. This improves and extends the blow up results in [16] by Liu and Wang (2006) and Gao et al. [7] (2011).

Article information

Source
Taiwanese J. Math., Volume 21, Number 4 (2017), 767-789.

Dates
Received: 29 July 2015
Revised: 13 October 2016
Accepted: 2 December 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501120834

Digital Object Identifier
doi:10.11650/tjm/6623

Mathematical Reviews number (MathSciNet)
MR3684386

Zentralblatt MATH identifier
06871345

Subjects
Primary: 35B44: Blow-up 35G61: Initial-boundary value problems for nonlinear higher-order systems 35L35: Initial-boundary value problems for higher-order hyperbolic equations 35L75: Nonlinear higher-order hyperbolic equations

Keywords
system of higher-order Kirchhoff-type equations nonlinear dissipation blow up lower bound of blow up time

Citation

Peyravi, Amir. Blow up Solutions to a System of Higher-order Kirchhoff-type Equations with Positive Initial Energy. Taiwanese J. Math. 21 (2017), no. 4, 767--789. doi:10.11650/tjm/6623. https://projecteuclid.org/euclid.twjm/1501120834


Export citation

References

  • R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York, 1975.
  • K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006), no. 11, 1235–1270.
  • J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 4, 473–486.
  • A. Benaissa and S. A. Messaoudi, Blow-up of solutions of a nonlinear wave equation, J. Appl. Math. 2 (2002), no. 2, 105–108.
  • ––––, Blow-up of solutions for the Kirchhoff equation of $q$-Laplacian type with nonlinear dissipation, Colloq. Math. 94 (2002), no. 1, 103–109.
  • W. Chen and Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal. 70 (2009), no. 9, 3203–3208.
  • Q. Gao, F. Li and Y. Wang, Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. Eur. J. Math. 9 (2011), no. 3, 686–698.
  • V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994), no. 2, 295–308.
  • A. Guesmia, Existence globale et stabilisation interne non linéarire d'un système de Petrovsky, Bell. Belg. Math. Soc. Simon Stevin 5 (1998), no. 4, 583–594.
  • R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal. 27 (1996), no. 10, 1165–1175.
  • V. Komornik, Well-posedness and decay estimates for a Petrovsky system by a semigroup approach, Acta. Sci. Math. (Szeged) 60 (1995), no. 3-4, 451–466.
  • H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_{tt} = -Au + \mc{F}(u)$, Trans. Amer. Math. Soc. 192 (1974), 1–21.
  • H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal. 137 (1997), no. 4, 341–361.
  • H. A. Levine and G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc. 129 (2001), no. 3, 793–805.
  • F. Li, Global existence and blow-up of solutions for a higher-order Kirchhoff-type equation with nonlinear dissipation, Appl. Math. Lett. 17 (2004), no. 2, 1409–1414.
  • L. Liu and M. Wang, Global existence and blow-up of solutions for some hyperbolic systems with damping and source terms, Nonlinear Anal. 64 (2006), no. 1, 69–91.
  • T. Matsuyama and R. Ikehata, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl. 204 (1996), no. 3, 729–753.
  • S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl. 265 (2002), no. 2, 296–308.
  • S. A. Messaoudi and B. Said Houari, A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation, Appl. Math. Lett. 20 (2007), no. 8, 866–871.
  • M. Ohta, Blowup of solutions of dissipative nonlinear wave equations, Hokkaido Math. J. 26 (1997), no. 1, 115–124.
  • K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations 137 (1997), no. 2, 273–301.
  • ––––, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl. 216 (1997), no. 1, 321–342.
  • ––––, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sei. 20 (1997), no. 2, 151–177.
  • J. Y. Park and J. J. Bae, On existence of solutions of degenerate wave equations with nonlinear damping terms, J. Korean Math. Soc. 35 (1998), no. 2, 465–490.
  • ––––, On existence of solutions of nondegenerate wave equations with nonlinear damping terms, Nihonkai Math. J. 9 (1998), no. 1, 27–46.
  • L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273–303.
  • S.-T. Wu, On coupled nonlinear wave equations of Kirchhoff type with damping and source terms, Taiwanese J. Math. 14 (2010), no. 2, 585–610.
  • S.-T. Wu and L.-Y. Tsai, On global existence and blow-up of solutions for an integro-differential equation with strong damping, Taiwanese J. Math. 10 (2006), no. 4, 979–1014.
  • ––––, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math. 13 (2009), no. 2A, 545–558.
  • E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999), no. 2, 155–182.
  • Y. Ye, Global existence and energy decay estimate of solutions for a higher-order Kirchhoff type equation with damping and source term, Nonlinear Anal. Real World Appl. 14 (2013), no. 6, 2059–2067
  • ––––, Global existence and asymptotic behavior of solutions for a system of higher-order Kirchhoff-type equations, Electron. J. Qual. Theory Differ. Equ. 2015 (2015), no. 20, 1–12.