## Journal of Applied Mathematics

### Blow-Up of Solutions for a Class of Sixth Order Nonlinear Strongly Damped Wave Equation

#### Abstract

We consider the blow-up phenomenon of sixth order nonlinear strongly damped wave equation. By using the concavity method, we prove a finite time blow-up result under assumptions on the nonlinear term and the initial data.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 310297, 6 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305767

Digital Object Identifier
doi:10.1155/2014/310297

Mathematical Reviews number (MathSciNet)
MR3212498

#### Citation

Di, Huafei; Shang, Yadong. Blow-Up of Solutions for a Class of Sixth Order Nonlinear Strongly Damped Wave Equation. J. Appl. Math. 2014 (2014), Article ID 310297, 6 pages. doi:10.1155/2014/310297. https://projecteuclid.org/euclid.jam/1425305767

#### References

• G. F. Webb, “Existence and asymptotic behavior for a strongly damped nonlinear wave equation,” Canadian Journal of Mathematics, vol. 32, no. 3, pp. 631–643, 1980.
• Y. C. Liu and D. C. Liu, “The initial-boundary value problem for the equation ${u}_{tt}-\alpha \Delta {u}_{t}-\Delta u=f(u)$,” Journal of Huazhong University of Science and Technology, vol. 16, no. 6, pp. 169–173, 1988.
• Y. D. Shang, “Blow-up of solutions for two classes of strongly damped nonlinear wave equations,” Journal of Engineering Mathematics, vol. 17, no. 2, pp. 65–70, 2000.
• F. X. Chen, B. L. Guo, and P. Wang, “Long time behavior of strongly damped nonlinear wave equations,” Journal of Differential Equations, vol. 147, no. 2, pp. 231–241, 1998.
• S. F. Zhou, “Attractors for strongly damped wave equation in uniform spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, pp. 174–187, 2006.
• A. B. Al'shin, M. O. Korpusov, and A. G. Sveshnikov, Blow-Up in Nonlinear Sobolev Type Equations, vol. 15 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2011.
• Y. D. Shang, “Initial-boundary value problem for the equation ${u}_{tt}-\Delta u-\Delta {u}_{t}-\Delta {u}_{tt}=f(u)$,” Acta Mathematicae Applicatae Sinica, vol. 23, no. 3, pp. 385–393, 2000.
• R. Z. Xu, S. Wang, Y. B. Yang, and Y. H. Ding, “Initial boundary value problem for a class of fourth-order wave equation with viscous damping term,” Applicable Analysis, vol. 92, no. 7, pp. 1403–1416, 2013.
• Y. C. Liu and R. Z. Xu, “Fourth order wave equations with nonlinear strain and source terms,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 585–607, 2007.
• G. Schneider and C. E. Wayne, “Kawahara dynamics in dispersive media,” Physica D: Nonlinear Phenomena, vol. 152-153, pp. 384–394, 2001.
• M. J. Boussinesq, Essai sur la Théorie des Eaux Courantes, vol. 3 of Mémoires Présentés par Divers Savans à l'Académie des Sciences de l'Institut de France (séries 2), 1877.
• Y. Wang and C. L. Mu, “Blow-up and scattering of solution for a generalized Boussinesq equation,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1131–1141, 2007.
• A. Esfahani, L. G. Farah, and H. Wang, “Global existence and blow-up for the generalized sixth-order Boussinesq equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 11, pp. 4325–4338, 2012.
• H. W. Wang and S. B. Wang, “Global existence and asymptotic behavior of solution for the Rosenau equation with hydrodynamical damped term,” Journal of Mathematical Analysis and Applications, vol. 401, no. 2, pp. 763–773, 2013.
• H. W. Wang and S. B. Wang, “Decay and scattering of small solutions for Rosenau equations,” Applied Mathematics and Computation, vol. 218, no. 1, pp. 115–123, 2011.
• L. C. Evance, Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1988.
• H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form $P{u}_{tt}=-Au+\mathcal{F}(u)$,” Transactions of the American Mathematical Society, vol. 192, pp. 1–21, 1974.
• J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, France, 1969.
• J. C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2001. \endinput