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Global existence and blowup of solutions for the multidimensional sixth-order “good” Boussinesq equation

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Abstract

This paper is concerned with the Cauchy problem of solutions for some nonlinear multidimensional “good” Boussinesq equation of sixth order at three different initial energy levels. In the framework of potential well, the global existence and blowup of solutions are obtained together with the concavity method at both low and critical initial energy level. Moreover by introducing a new stable set, we present some sufficient conditions on initial data such that the weak solution exists globally at supercritical initial energy level.

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References

  1. Boussinesq J.V.: Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal. C. R. Acad. Sci. Paris 73, 256–260 (1871)

    MATH  Google Scholar 

  2. Boussinesq J.V.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)

    MATH  Google Scholar 

  3. Angulo, P.J., Scialom, M.: Improved blow-up of solution of a generalized Boussinesq equation. Comput. Appl. Math. 18(3), 333–341, 371 (1999)

  4. Bona J.L., Sachs R.L.: Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun. Math. Phys. 118(1), 15–29 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Farah L.G.: Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Commun. Pure Appl. Anal. 08, 1521–1539 (2009)

    Article  MathSciNet  Google Scholar 

  6. Linares F.: Global existence of small solutions for a generalized Boussinesq equation. J. Differ. Equ. 106(2), 257–293 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Liu Y.: Instability and blow-up of solutions to a generalized Boussinesq equation. SIAM J. Math. Anal. 26(6), 1527–1546 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu Y.: Decay and scattering of small solutions of a generalized Boussinesq equation. J. Funct. Anal. 147(1), 51–61 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Schneidera G., Wayne C.E.: Kawahara dynamics in dispersive media. Phys. D Nonlinear Phenom. 152-153, 384–394 (2001)

    Article  Google Scholar 

  10. Boussinesq M.J.: Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants á I, Académie des Sciences Inst. France, Séries 2(3), 1–680 (1877)

    Google Scholar 

  11. Wang S., Xue H.: Global solution for a generalized Boussinesq equation. Appl. Math. Comput. 204, 130–136 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wang Y., Mu C., Wu Y.: Decay and scattering of solutions for a generalized Boussinesq equation. J. Differ. Equ. 247, 2380–2394 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wang Y., Guo B.: The Cauchy problem for a generalized Boussinesq type equation. Chin. J. Contemp. Math. 29((2), 185–194 (2008)

    Google Scholar 

  14. Wang Y., Mu C.: Blow-up and scattering of solution for a generalized Boussinesq equation. Appl. Math. Comput. 188, 1131–1141 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wang Y., Mu C., Li N.: Scattering of small amplitude solutions for a multidimensional generalized Boussinesq equation. Dyn. Contin. Discret. Impuls. Syst. A 14, 593–607 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Wang Y., Mu C.: Global existence and blow-up of the solutions for the multidimensional generalized Boussinesq equation. Math. Methods Appl. Sci. 30, 1403–1417 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Xia S., Yuan J.: Existence and scattering of small solutions to a Boussinesq type equation of sixth order. Nonlinear Anal. 73, 1015–1027 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Taskesen, H., Polat, N., Erta, A.: On global solutions for the Cauchy Problem of a Boussinesq-type equation. Abstract and Applied Analysis, volume 2012 (2012). Article ID 535031, 10 pp.

  19. Taskesen, H., Polat, N.: Existence of global solutions for a multidimensional Boussinesq-type equation with supercritical initial energy. In: First International Conference on Analysis and Applied Mathematics: ICAAM 2012, AIP Conference Proceedings, 1470, pp. 159–162

  20. Kutev N., Kolkovska N., Dimova M.: Global existence of Cauchy problem for Boussinesq paradigm equation. Comput. Math. Appl. 65, 500–511 (2013)

    Article  MathSciNet  Google Scholar 

  21. Xu R., Liu Y., Liu B.: Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations. Nonlinear Anal. 74, 2425–2437 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Payne L.E., Sattinger D.H.: Saddle points and instability on nonlinear hyperbolic equations. Israel J. Math. 22, 273–303 (1975)

    Article  MathSciNet  Google Scholar 

  23. Levine H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  24. Levine H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = −AuF(u). Trans. Am. Math. Soc. 192, 1–21 (1974)

    MATH  Google Scholar 

  25. Liu Y., Xu R.: Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation. Phys. D Nonlinear Phenom. 237(6), 721–731 (2008)

    Article  MATH  Google Scholar 

  26. Xu R.: Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Q. Appl. Math. 68, 459–468 (2010)

    Article  MATH  Google Scholar 

  27. Yacheng L., Junsheng Z.: On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. 64, 2665–2687 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Wang, S., Yang, Y., Xu, R.: Finite time blow up of solution of Cauchy problem for the multidimensional generalized double dispersion equations with arbitrary initial energy. International Conference on Multimedia Technology, ICMT 2011, pp. 2118–2121 (2011)

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Runzhang, X., Yanbing, Y., Bowei, L. et al. Global existence and blowup of solutions for the multidimensional sixth-order “good” Boussinesq equation. Z. Angew. Math. Phys. 66, 955–976 (2015). https://doi.org/10.1007/s00033-014-0459-9

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