Differential Equations and Nonlinear Mechanics

Global Existence and Asymptotic Behavior of Solutions for a Class of Nonlinear Degenerate Wave Equations

Yaojun Ye

Full-text: Open access

Abstract

This paper studies the existence of global solutions to the initial-boundary value problem for some nonlinear degenerate wave equations by means of compactness method and the potential well idea. Meanwhile, we investigate the decay estimate of the energy of the global solutions to this problem by using a difference inequality.

Article information

Source
Differ. Equ. Nonlinear Mech., Volume 2007 (2007), Article ID 019685, 9 pages.

Dates
Received: 20 December 2006
Accepted: 10 April 2007
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399782

Digital Object Identifier
doi:10.1155/2007/19685

Mathematical Reviews number (MathSciNet)
MR2318209

Zentralblatt MATH identifier
1138.35376

Citation

Ye, Yaojun. Global Existence and Asymptotic Behavior of Solutions for a Class of Nonlinear Degenerate Wave Equations. Differ. Equ. Nonlinear Mech. 2007 (2007), Article ID 019685, 9 pages. doi:10.1155/2007/19685. https://projecteuclid.org/euclid.ijde/1485399782


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References

  • M. Aassila, “Global existence and global nonexistence of solutions to a wave equation with nonlinear damping and source terms,” Asymptotic Analysis, vol. 30, no. 3-4, pp. 301–311, 2002. MR1932035
  • V. Georgiev and G. Todorova, “Existence of a solution of the wave equation with nonlinear damping and source terms,” Journal of Differential Equations, vol. 109, no. 2, pp. 295–308, 1994. MR1273304
  • M. Nakao and K. Ono, “Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,” Mathematische Zeitschrift, vol. 214, no. 2, pp. 325–342, 1993. MR1240892
  • G. Todorova, “Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms,” Journal of Mathematical Analysis and Applications, vol. 239, no. 2, pp. 213–226, 1999. MR1723057
  • J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, France, 1969. MR0259693
  • D. H. Sattinger, “On global solution of nonlinear hyperbolic equations,” Archive for Rational Mechanics and Analysis, vol. 30, pp. 148–172, 1968. MR0227616
  • M. Nakao, “A difference inequality and its application to nonlinear evolution equations,” Journal of the Mathematical Society of Japan, vol. 30, no. 4, pp. 747–762, 1978. MR513082
  • Y. Ye, “Existence of global solutions for some nonlinear hyperbolic equation with a nonlinear dissipative term,” Journal of Zhengzhou University. Natural Science Edition, vol. 29, no. 3, pp. 18–23, 1997. MR1620772