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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 760209, 14 pages
doi:10.1155/2011/760209
Research Article
Global Nonexistence of Positive Initial-Energy Solutions for Coupled Nonlinear Wave Equations with Damping and Source Terms
1Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Science, Nanjing Normal University, Nanjing 210046, China
2Department of Mathematics, Anhui Science and Technology University, Fengyang 233100, Anhui, China
Received 26 November 2010; Revised 5 June 2011; Accepted 27 June 2011
Academic Editor: Josef Diblík
Copyright © 2011 Liang Fei and Gao Hongjun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- A. Haraux and E. Zuazua, “Decay estimates for some semilinear damped hyperbolic problems,” Archive for Rational Mechanics and Analysis, vol. 100, no. 2, pp. 191–206, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. M. Ball, “Remarks on blow-up and nonexistence theorems for nonlinear evolution equations,” The Quarterly Journal of Mathematics. Oxford. Second Series, vol. 28, no. 112, pp. 473–486, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- V. K. Kalantarov, “The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types,” Journal of Soviet Mathematics, vol. 10, no. 1, pp. 53–70, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form ,” Transactions of the American Mathematical Society, vol. 192, pp. 1–21, 1974. View at Zentralblatt MATH
- H. A. Levine, “Some additional remarks on the nonexistence of global solutions to nonlinear wave equations,” SIAM Journal on Mathematical Analysis, vol. 5, pp. 138–146, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S. A. Messaoudi, “Blow up in a nonlinearly damped wave equation,” Mathematische Nachrichten, vol. 231, pp. 105–111, 2001. View at Zentralblatt MATH
- S. A. Messaoudi, “Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 902–915, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- G. Andrews, “On the existence of solutions to the equation ,” Journal of Differential Equations, vol. 35, no. 2, pp. 200–231, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- G. Andrews and J. M. Ball, “Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity,” Journal of Differential Equations, vol. 44, no. 2, pp. 306–341, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Clements, “Existence theorems for a quasilinear evolution equation,” SIAM Journal on Applied Mathematics, vol. 26, pp. 745–752, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- S. Kawashima and Y. Shibata, “Global existence and exponential stability of small solutions to nonlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- D. D Ang and A. P. N. Dinh, “Strong solutions of a quasilinear wave equation with nonlinear damping,” SIAM Journal on Mathematical Analysis, vol. 19, no. 2, pp. 337–347, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
- W. N. Findley, J. S. Lai, and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland, Amsterdam, The Netherlands, 1976.
- J. K. Knowles, “On finite anti-plane shear for incompressible elastic materials,” Australian Mathematical Society. Journal. Series B, vol. 19, no. 4, pp. 400–415, 1975/76.
- V. P. Maslov and P. P. Mosolov, Nonlinear Wave Equations Perturbed by Viscous Terms, vol. 31 of de Gruyter Expositions in Mathematics, Walter De Gruyter, Berlin, Germany, 2000.
- J. M. Greenberg, R. C. MacCamy, and V. J. Mizel, “On the existence, uniqueness, and stability of solutions of the equation ,” vol. 17, pp. 707–728, 1967/1968.
- Y. Yamada, “Some remarks on the equation ,” Osaka Journal of Mathematics, vol. 17, no. 2, pp. 303–323, 1980. View at Zentralblatt MATH
- Z. Yang and G. Chen, “Global existence of solutions for quasi-linear wave equations with viscous damping,” Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 604–618, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- G. Chen, H. Yue, and S. Wang, “The initial boundary value problem for quasi-linear wave equation with viscous damping,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 823–839, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Hao, Y. Zhang, and S. Li, “Global existence and blow-up phenomena for a nonlinear wave equation,” Nonlinear Analysis, vol. 71, no. 10, pp. 4823–4832, 2009. View at Publisher · View at Google Scholar
- S. A. Messaoudi and B. Said Houari, “Global non-existence of solutions of a class of wave equations with non-linear damping and source terms,” Mathematical Methods in the Applied Sciences, vol. 27, no. 14, pp. 1687–1696, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- I. E. Segal, “Nonlinear partial differential equations in quantum field theory,” in Proc. Sympos. Appl. Math., Vol. XVII, pp. 210–226, American Mathematical Society, Providence, RI, USA, 1965. View at Zentralblatt MATH
- K. Jorgens, Nonlinear Wave Equations, University of Colorado, Department of Mathematics, 1970.
- K. Agre and M. A. Rammaha, “Systems of nonlinear wave equations with damping and source terms,” Differential and Integral Equations, vol. 19, no. 11, pp. 1235–1270, 2006. View at Zentralblatt MATH
- 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 Equations, vol. 23, no. 1-2, pp. 79–92, 2010. View at Zentralblatt MATH
- J. Wu, S. Li, and S. Chai, “Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source,” Nonlinear Analysis, vol. 72, no. 11, pp. 3969–3975, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- L. E. Payne and D. H. Sattinger, “Saddle points and instability of nonlinear hyperbolic equations,” Israel Journal of Mathematics, vol. 22, no. 3-4, pp. 273–303, 1975. View at Publisher · View at Google Scholar · View at MathSciNet
- E. Vitillaro, “Global nonexistence theorems for a class of evolution equations with dissipation,” Archive for Rational Mechanics and Analysis, vol. 149, no. 2, pp. 155–182, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- S. A. Messaoudi and B. Said-Houari, “Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 277–287, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH