`Abstract and Applied AnalysisVolume 2012, Article ID 615345, 27 pageshttp://dx.doi.org/10.1155/2012/615345`
Research Article

## On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source

Department of Mathematics, Southeast University, Nanjing 210018, China

Received 24 May 2012; Revised 17 July 2012; Accepted 17 July 2012

Copyright © 2012 Xiaopan Liu. 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.

1. J. M. Greenberg, R. C. MacCamy, and V. J. Mizel, “On the existence,uniqueness and stability of solutions of the equation ${\sigma }^{\prime }\left({u}_{xx}\right)+\lambda {u}_{xxt}={\rho }_{0}{u}_{tt}$,” Journal of Mathematics and Mechanics, vol. 17, pp. 707–728, 1968.
2. J. Clements, “Existence theorems for a quasilinear evolution equation,” SIAM Journal on Applied Mathematics, vol. 26, pp. 745–752, 1974.
3. J. C. Clements, “On the existence and uniqueness of solutions of the equation ${u}_{tt}-\left(\partial /{\partial }_{xi}\right){\sigma }_{i}\left({u}_{xi}\right)-{\mathrm{\Delta }}_{N}{u}_{t}=f$,” Canadian Mathematical Bulletin, vol. 18, no. 2, pp. 181–187, 1975.
4. Z. Yang, “Global existence, asymptotic behavior and blow-up of solutions for a class of nonlinear wave equations with dissipative term,” Journal of Differential Equations, vol. 187, no. 2, pp. 520–540, 2003.
5. 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.
6. F. Sun and M. Wang, “Non-existence of global solutions for nonlinear strongly damped hyperbolic systems,” Discrete and Continuous Dynamical Systems Series A, vol. 12, no. 5, pp. 949–958, 2005.
7. F. Sun and M. Wang, “Global and blow-up solutions for a system of nonlinear hyperbolic equations with dissipative terms,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 4, pp. 739–761, 2006.
8. S. Berrimi and S. A. Messaoudi, “Existence and decay of solutions of a viscoelastic equation with a nonlinear source,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 10, pp. 2314–2331, 2006.
9. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping,” Electronic Journal of Differential Equations, vol. 44, pp. 1–14, 2002.
10. X. Han and M. Wang, “General decay of energy for a viscoelastic equation with nonlinear damping,” Mathematical Methods in the Applied Sciences, vol. 32, no. 3, pp. 346–358, 2009.
11. G. Li, Y. Sun, and W. Liu, “Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping,” Applicable Analysis, vol. 91, no. 3, pp. 575–586, 2012.
12. W. Liu, “Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source,” Topological Methods in Nonlinear Analysis, vol. 36, no. 1, pp. 153–178, 2010.
13. W. Liu, “Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms,” Frontiers of Mathematics in China, vol. 5, no. 3, pp. 555–574, 2010.
14. C. Song and Z. Yang, “Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation,” Mathematical Methods in the Applied Sciences, vol. 33, no. 5, pp. 563–575, 2010.
15. R. Xu, Y. Liu, and T. Yu, “Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4977–4983, 2009.