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http://dx.doi.org/10.4134/BKMS.b170634

GLOBAL SOLUTIONS FOR A CLASS OF NONLINEAR SIXTH-ORDER WAVE EQUATION  

Wang, Ying (School of Mathematical Sciences University of Electronic Science and Technology of China)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.4, 2018 , pp. 1161-1178 More about this Journal
Abstract
In this paper, we consider the Cauchy problem for a class of nonlinear sixth-order wave equation. The global existence and the finite time blow-up for the problem are proved by the potential well method at both low and critical initial energy levels. Furthermore, we present some sufficient conditions on initial data such that the weak solution exists globally at supercritical initial energy level by introducing a new stable set.
Keywords
Cauchy problem; sixth-order wave equation; blow-up; existence of global solution;
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1 H. Taskesen, N. Polat, and A. Erta, On global solutions for the Cauchy problem of a Boussinesq-type equation, Abstr. Appl. Anal. (2012), Art. ID 535031, 10 pp.
2 Y. Wang and C. Mu, Blow-up and scattering of solution for a generalized Boussinesq equation, Appl. Math. Comput. 188 (2007), no. 2, 1131-1141.   DOI
3 Y. Wang and C. Mu, Global existence and blow-up of the solutions for the multidimensional generalized Boussinesq equation, Math. Methods Appl. Sci. 30 (2007), no. 12, 1403-1417.   DOI
4 Y. Wang, C. Mu and J. Deng, Strong instability of solitary-wave solutions for a nonlinear Boussinesq equation, Nonlinear Anal. 69 (2008), no. 5-6, 1599-1614.   DOI
5 Y.-Z. Wang and Y.-X. Wang, Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order, Nonlinear Anal. 72 (2010), no. 12, 4500- 4507.   DOI
6 R. Z. Xu, Y. B. Yang, B. W. Liu, J. H. Shen, and S. B. Huang, Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys. 66 (2015), no. 3, 955-976.   DOI
7 A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differential Integral Equations 15 (2002), no. 9, 1061-1072.
8 G. B. Folland, Real Analysis, second edition, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999.
9 T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907.   DOI
10 N. Kutev, N. Kolkovska, and M. Dimova, Global existence of Cauchy problem for Boussinesq paradigm equation, Comput. Math. Appl. 65 (2013), no. 3, 500-511.   DOI
11 S. Lai and Y. Wu, The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation, Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 3, 401-408.   DOI
12 Q. Lin, Y. H. Wu, and R. Loxton, On the Cauchy problem for a generalized Boussinesq equation, J. Math. Anal. Appl. 353 (2009), no. 1, 186-195.   DOI
13 Y. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations 192 (2003), 109-127.
14 T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, 3, Walter de Gruyter & Co., Berlin, 1996.
15 Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D 237 (2008), no. 6, 721-731.   DOI
16 N. Polat and A. Ertas, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl. 349 (2009), no. 1, 10-20.   DOI
17 N. Polat and E. Piskin, Asymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation, Appl. Math. Lett. 25 (2012), no. 11, 1871-1874.   DOI
18 G. Schneider and C. E. Wayne, Kawahara dynamics in dispersive media, Phys. D 152/153 (2001), 384-394.   DOI
19 H. Wang and A. Esfahani, Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation, Nonlinear Anal. 89 (2013), 267-275.   DOI
20 H. Wang and A. Esfahani, Global rough solutions to the sixth-order Boussinesq equation, Nonlinear Anal. 102 (2014), 97-104.   DOI
21 H. Taskesen and N. Polat, Existence of global solutions for a multidimensinal Boussinesq type equation with supercritical initial energy, In: First international Conference on Analysis and Applied Mathematics: ICAAM, AIP Conference Proceedings 1470 (2012), 159-162.
22 S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. 64 (2006), no. 1, 159-173.   DOI
23 S. Wang and G. Xu, The Cauchy problem for the Rosenau equation, Nonlinear Anal. 71 (2009), no. 1-2, 456-466.   DOI
24 S. Wang and H. Xue, Global solution for a generalized Boussinesq equation, Appl. Math. Comput. 204 (2008), no. 1, 130-136.   DOI