## Abstract and Applied Analysis

### On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

#### Abstract

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity $\gamma {|u|}^{p}$ and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 535031, 10 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365174054

Digital Object Identifier
doi:10.1155/2012/535031

Mathematical Reviews number (MathSciNet)
MR2947731

Zentralblatt MATH identifier
1246.35184

#### Citation

Taskesen, Hatice; Polat, Necat; Ertaş, Abdulkadir. On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 535031, 10 pages. doi:10.1155/2012/535031. https://projecteuclid.org/euclid.aaa/1365174054

#### References

• N. Polat and A. Ertaş, “Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation,” Journal of Mathematical Analysis and Applications, vol. 349, no. 1, pp. 10–20, 2009.
• R. Xue, “Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 307–327, 2006.
• N. Polat, “Existence and blow up of solutions of the Cauchy problem of the generalized damped multidimensional improved modified Boussinesq equation,” Zeitschrift Für Naturforschung A, vol. 63, pp. 1–10, 2008.
• Q. Lin, Y. H. Wu, and S. Lai, “On global solution of an initial boundary value problem for a class of damped nonlinear equations,” Nonlinear Analysis, vol. 69, no. 12, pp. 4340–4351, 2008.
• N. Polat and D. Kaya, “Blow up of solutions for the generalized Boussinesq equation with damping term,” Zeitschrift Für Naturforschung A, vol. 61, pp. 235–238, 2006.
• Y. Liu and R. Xu, “Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation,” Physica D, vol. 237, no. 6, pp. 721–731, 2008.
• Q. Lin, Y. H. Wu, and R. Loxton, “On the Cauchy problem for a generalized Boussinesq equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 186–195, 2009.
• X. Runzhang, “Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities,” Mathematical Methods in the Applied Sciences, vol. 34, no. 18, pp. 2318–2328, 2011.
• J. A. Esquivel-Avila, “Dynamics around the ground state of a nonlinear evolution equation,” Nonlinear Analysis, vol. 63, no. 5–7, pp. 331–343, 2005.
• R. Xu, Y. Liu, and B. Liu, “The Cauchy problem for a class of the multidimensional Boussinesq-type equation,” Nonlinear Analysis, vol. 74, no. 6, pp. 2425–2437, 2011.
• Y.-Z. Wang and Y.-X. Wang, “Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order,” Nonlinear Analysis, vol. 72, no. 12, pp. 4500–4507, 2010.
• S. Wang and G. Xu, “The Cauchy problem for the Rosenau equation,” Nonlinear Analysis, vol. 71, no. 1-2, pp. 456–466, 2009.
• N. Kutev, N. Kolkovska, M. Dimova, and C. I. Christov, “Theoretical and numerical aspects for global existence and blow up for the solutions to Boussinesq paradigm equation,” AIP Conference Proceedings, vol. 1404, pp. 68–76, 2011.
• E. H. Lieb, “Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,” Annals of Mathematics, vol. 118, no. 2, pp. 349–374, 1983.
• H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu=A{u}_{tt}+F(u)$,” Transactions of the American Mathematical Society, vol. 192, pp. 1–21, 1974.