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Global existence and blow up for damped generalized Boussinesq equation

  • Run-zhang XuEmail author
  • Yong-bing Luo
  • Ji-hong Shen
  • Shao-bin Huang
Article

Abstract

We study the Cauchy problem of damped generalized Boussinesq equation u tt u xx + (u xx + f(u)) xx αu xxt = 0. First we give the local existence of weak solution and smooth solution. Then by using potential well method and convexity method we prove the global existence and finite time blow up of solution, then we obtain some sharp conditions for the well-posedness problem.

Keywords

generalized Boussinesq equation damping Cauchy problem global existence blow up 

2000 MR Subject Classification

35Q35 

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Notes

Acknowledgements

We do appreciate the referee’s so many valuable suggestions, which corrected some mistakes in the paper and improved the paper a lot.

References

  1. [1]
    Bona, J., Sachs, R. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm. Math. Phys., 118: 15–29 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Deift, P., Tomei, C., Trubowitz, E. Inverse scattering and the Boussinesq equation. Comm. Pure Appl. Math., 35: 567–628 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Galkin, V.M., Pelinovsky, D.E., Stepanyants, Y.A. The structure of the rational solutions to the Boussinesq equation. Physica D, 80: 246–255 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Hirota, R. Solutions of the classical Boussinesq equation and the spherical Boussinesq equation: the Wronskian technique. J. Phys. Soc. Japan, 55: 2137–2150 (1986)CrossRefGoogle Scholar
  5. [5]
    Levine, H.A. Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = −Au + F(u). Trans. Amer. Math. Soc., 192: 1–21 (1974)MathSciNetGoogle Scholar
  6. [6]
    Levine, H.A. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal., 5: 138–146 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Linares, F. Global existence of small solutions for a generalized Boussinesq equation. Journal of Differential Equations, 106: 257–293 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Linares, F., Scialom, M. Asymptotic behavior of solutions of a generalized Boussinesq-type equation. Nonlinear Anal. TMA, 25: 1147–1158 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Lin, Q., Wu, Y.H., Loxton, R. On the Cauchy problem for a generalized Boussinesq equation. Journal of Mathematical Analysis and Applications, 353: 186–195 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Lin, Q., Wu, Y.H., Lai, S. On global solution of an initial boundary value problem for a class of damped nonlinear equations. Nonlinear Anal. TMA, 69: 4340–4351 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Liu, Y. Instability and blow-up of solutions to a generalized Boussinesq equation. SIAM J. Math. Anal., 26: 1527–1546 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Liu, Y., Xu, R. Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation. Physica D: Nonlinear Phenomena, 237: 721–731 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Liu, Y. Instability of solitary waves for generalized Boussinesq equations. J. Dynamics Differential Equations, 537–558 (1993)Google Scholar
  14. [14]
    Pego, R.L., Weinstein, M.I. Eigenvalues and instabilities of solitary waves, Philos. Trans. Roy. Soc. London A, 340: 47–94 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Tsutsumi, M., Matahashi, T. On the Cauchy problem for the Boussinesq-type equation. Math. Japan, 36: 371–379 (1991)MathSciNetzbMATHGoogle Scholar
  16. [16]
    Varlamov, V.V. On the Cauchy problem for the damped Boussinesq equation. Differential Integral Equations, 9(3): 619–634 (1996)MathSciNetzbMATHGoogle Scholar
  17. [17]
    Varlamov, V.V. On spatially periodic solutions of the damped Boussinesq equation. Differential Integral Equations, 10(6): 1197–1211 (1997)MathSciNetzbMATHGoogle Scholar
  18. [18]
    Varlamov, V.V. On the initial-boundary value problem for the damped Boussinesq equation. Discrete Continuous Dyn. Systems, 4(3): 431–444 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Varlamov, V.V. Long-time asymptotics of solutions of the damped Boussinesq equation. Abstract Appl. Anal., 2(3/4): 97–115 (1998)Google Scholar
  20. [20]
    Varlamov, V.V. Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions. Internat. J. Maths. Math. Sci., 22(1): 131–145 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Varlamov, V.V. On the damped Boussinesq equation in a circle. Nonlinear Anal. TMA, 38: 447–470 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Xue, R. Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation. J. Math. Anal. Appl., 316: 307–327 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Run-zhang Xu
    • 1
    Email author
  • Yong-bing Luo
    • 2
  • Ji-hong Shen
    • 1
  • Shao-bin Huang
    • 3
  1. 1.College of ScienceHarbin Engineering UniversityHarbinChina
  2. 2.College of AutomationHarbin Engineering UniversityHarbinChina
  3. 3.College of Computer Science and TechnologyHarbin Engineering UniversityHarbinChina

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