Skip to main content
SpringerLink
Log in

Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The solvability of the natural (first, second, and mixed) initial boundary-value problems for nonlinear analogs of the Boussinesq equation is studied. Uniqueness theorems for regular solutions and global solvability theorems are proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. B. Whitham, Linear and NonlinearWaves (Wiley-Interscience, New York–London–Sydney, 1974; Mir, Moscow, 1977).

    Google Scholar 

  2. H. Ikezi, “Experiments on solitons in plasmas,” in Solitons in Action, Proceedings of a Workshop held at Redstone Arsenal, Ala., October 26–27, 1977 (Redstone Arsenal, Ala., 1977; Mir, Moscow, 1981), pp. 153–172.

    Google Scholar 

  3. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic Press, London, 1982; Mir, Moscow, 1988).

    MATH  Google Scholar 

  4. G. V. Demidenko and S. V. Uspenskii, Equations and Systems that Are Not Solved with Respect to the Highest Derivative (Nauchn. Kniga, Novosibirsk, 1998) [in Russian].

    MATH  Google Scholar 

  5. G. Chen and S. Wang, “Existence and nonexistence of global solutions for the generalized IMBq equation,” Nonlinear Anal., TheoryMethods Appl. 36 (8), 961–980 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Wang and G. Chen, “Small amplutude solutions of the generalized IMBq equation,” J. Math. Anal. Appl. 274 (2), 846–866 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Xu and Y. Liu, “Global existence and blow-up of solutions for generalized Pochhammer–Chree equations,” Acta Math. Sci. Ser. B Engl. Ed. 30 (5), 1793–1807 (2010).

    MathSciNet  MATH  Google Scholar 

  8. A. A. Zamyshlyaeva and A. V. Yuzeeva, “The initial-terminal value problem for the Boussinesq–Love equation,” Vestnik YuUrGU Ser.Mat. Model. Programm., No. 5, 23–31 (2010).

    MATH  Google Scholar 

  9. A. A. Zamyshlyaeva, “The initial-terminal value problem for the inhomogeneous Boussinesq–Love equation,” Vestnik YuUrGU Ser.Mat.Model. Programm., No. 10, 22–29 (2011).

    Google Scholar 

  10. E. A. Utkina, “A uniqueness theorem for the solution of a Dirichlet problem,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 62–67 (2011) [Russian Math. (Iz. VUZ) 55, No. 5, 51–55 (2011)].

    MathSciNet  MATH  Google Scholar 

  11. E. A. Utkina, “A boundary-value problem with conditions on the entire boundary of a noncharacteristic domain for a fourth-order equation,” Dokl. Ross. Akad. Nauk 439 (4), 597–599 (2011) [Dokl. Math. 84 (1), 539–541 (2011)].

    MathSciNet  MATH  Google Scholar 

  12. A. A. Zamyshlyaeva and O. N. Tsyplenkova, “Optimal control of the solutions of the initial-terminal value problem for the Boussinesq–Love equation,” Vestnik YuUrGU Ser.Mat.Model. Programm., No. 11, 13–24 (2012).

    MATH  Google Scholar 

  13. M. V. Plekhanova and A. F. Islamova, “Problems with a robust mixed control for the linearized Boussinesq equation,” Differ. Uravn. 48 (4), 565–576 (2012). [Differ. Equations 48 (4), 574–585 (2012).]

    MathSciNet  MATH  Google Scholar 

  14. L. S. Pul’kina, Problems with Nonclassical Conditions for Hyperbolic Equations (Izd. “Samarskii Univ.,” Samara, 2012) [in Russian].

    MATH  Google Scholar 

  15. Karl E. Lonngren, “Observations of solitons on nonlinear dispersive transmission lines,” in Solitons in Action, Proceedings of aWorkshop held at Redstone Arsenal, Ala., October 26–27, 1977 (Redstone Arsenal, Ala., 1977; Mir, Moscow, (1981), pp. 127–152.

    Google Scholar 

  16. B. P. Demidovich, Lectures in Mathematical Stability Theory (Nauka, Moscow, 1967) [in Russian].

    MATH  Google Scholar 

  17. L. Bers, F. John, and M. Schechter, Partial Differential Equations (Interscience, New York, 1964; Mir, Moscow, 1966).

    MATH  Google Scholar 

  18. N. A. Lar’kin, V. A. Novikov, and N. N. Yanenko, Nonlinear Equations of Variable Type (Nauka, Novosibirsk, 1983) [in Russian].

    MATH  Google Scholar 

  19. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  20. O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Qusilinear Equations of Elliptic Type (Nauka, Moscow, 1973) [in Russian].

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Karabük University, Karabük, Turkey

    Sh. Amirov

  2. Sobolev Institute of Mathematics, Siberia Division of Russian Academy of Sciences, Novosibirsk, Russia

    A. I. Kozhanov

  3. Novosibirsk National Research State University, Novosibirsk, Russia

    A. I. Kozhanov

Authors
  1. Sh. Amirov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. I. Kozhanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Kozhanov.

Additional information

Original Russian Text © Sh. Amirov, A. I. Kozhanov, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 2, pp. 171–180.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amirov, S., Kozhanov, A.I. Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation. Math Notes 99, 183–191 (2016). https://doi.org/10.1134/S0001434616010211

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434616010211

Keywords

Search

Navigation