- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 760209, 14 pages
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- S. A. Messaoudi, “Blow up in a nonlinearly damped wave equation,” Mathematische Nachrichten, vol. 231, pp. 105–111, 2001.
- 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.
- G. Andrews, “On the existence of solutions to the equation ,” Journal of Differential Equations, vol. 35, no. 2, pp. 200–231, 1980.
- 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.
- J. Clements, “Existence theorems for a quasilinear evolution equation,” SIAM Journal on Applied Mathematics, vol. 26, pp. 745–752, 1974.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.