Skip to main content
SpringerLink
Log in

Mixed problem for the Kawahara equation in a half-strip

  • Partial Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We prove the global well-posedness of the mixed problem for the Kawahara equation in a half-strip under natural conditions on the boundary data.

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. Kawahara, T., Oscillatory Solitary Waves in Dispersive Media, J. Phys. Soc. Japan, 1972, vol. 33, no. 1, pp. 260–264.

    Article  MathSciNet  Google Scholar 

  2. Marchenko, A.V., On Long Waves in a Shallow Liquid Under an Ice Cover, Prikl. Mat. Mekh., 1988, vol. 52, no. 2, pp. 230–234.

    Google Scholar 

  3. Il’ichev, A.T., Properties of a Fifth-Order Nonlinear Evolution Equation That Describes Wave Processes in Media with Weak Dispersion, Tr. Mat. Inst. Steklova, 1989, vol. 186, pp. 222–226.

    MathSciNet  Google Scholar 

  4. Pomeau, Y., Ramani, A., and Grammaticos, B., Structural Stability of the Korteweg-de Vries Solitons Under a Singular Perturbation, Physica D, 1988, vol. 31, pp. 127–134.

    Article  MATH  MathSciNet  Google Scholar 

  5. Boyd, J.P., Weakly Non-Local Solitons for Capillary-Gravity Waves: Fifth Degree Korteweg-de Vries Equation, Physica D, 1991, vol. 48, pp. 129–146.

    Article  MATH  Google Scholar 

  6. Saut, J.C., Sur quelques généralizations de l’equation de Korteweg-de Vries, J. Math. Pures Appl. (9), 1979, vol. 58, no. 1, pp. 21–61.

    MATH  MathSciNet  Google Scholar 

  7. Faminskii, A.V., The Cauchy Problem for Quasilinear Equations of Odd Order, Mat. Sb., 1989, vol. 180, no. 9, pp. 1183–1210.

    Google Scholar 

  8. Biagioni, H.A. and Linares, F., On the Benney-Lin and Kawahara Equations, J. Math. Anal. Appl., 1997, vol. 211, pp. 131–152.

    Article  MATH  MathSciNet  Google Scholar 

  9. Cui, S. and Tao, S., Stricharts Estimates for Dispersive Equations and Solvability of the Kawahara Equation, J. Math. Anal. Appl., 2005, vol. 304, pp. 683–702.

    Article  MATH  MathSciNet  Google Scholar 

  10. Cui, S., Deng, D., and Tao, S., Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L 2 Initial Data, Acta Math. Sin. (Engl. Ser.), 2006. vol. 22, no. 5, pp. 1457–1466.

    Article  MATH  MathSciNet  Google Scholar 

  11. Wang, H., Cui, S., and Deng, D., Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices, Acta Math. Sin. (Engl. Ser.), 2007, vol. 23, no. 8, pp. 1435–1446.

    Article  MATH  MathSciNet  Google Scholar 

  12. Sangare, K., Mixed Problem in a Half-Strip for a Generalized Kawahara Equation in the Space of Infinitely Differentiable Exponentially Decreasing Functions, Vestn. Ross. Univ. Druzhby Narodov. Ser. Mat., 2003, vol. 10, no. 1, pp. 91–107.

    Google Scholar 

  13. Larkin, N.A. and Doronin, G.G., Kawahara Equation in a Quarter-Plane and in a Finite Domain, Bol. Soc. Parana. Mat. (3), 2007, vol. 25, no. 1–2, pp. 9–16.

    MATH  MathSciNet  Google Scholar 

  14. Doronin, G.G. and Larkin, N.A., Boundary Value Problems for the Stationary Kawahara Equation, Nonlinear Analysis Series A: Theory, Methods & Appl., 2007. doi: 10.1016/j.na.200707005.

  15. Faminskii, A.V., An Initial Boundary-Value Problem in a Half-Strip for the Korteweg-de Vries Equation in Fractional-Order Sobolev Spaces, Comm. Partial Differential Equations, 2004, vol. 29, no. 11–12, pp. 1653–1695.

    MATH  MathSciNet  Google Scholar 

  16. Faminskii, A.V., Global Well-Posedness of Two Initial-Boundary-Value Problems for the Korteweg-de Vries Equation, Differ. Integral Eq. Appl., 2007, vol. 20, no. 6, pp. 601–642.

    MathSciNet  Google Scholar 

  17. Kenig, C.E., Ponce, G., and Vega, L., Well-Posedness of the Initial Value Problem for the Korteweg-de Vries Equation, J. Amer. Math. Soc., 1991, vol. 4, no. 2, pp. 323–347.

    Article  MATH  MathSciNet  Google Scholar 

  18. Lions, J.-L. and Magenes, E., Problèmes aux limites non homogénes et applications, Paris: Dunod, 1968. Translated under the title Neodnorodnye granichnye zadachi i ikh prilozheniya, Moscow: Mir, 1971.

    MATH  Google Scholar 

  19. Bona, J.L., Sun, S., and Zhang, B.-Y., A Nonhomogeneous Boundary-Value Problems for the Korteweg-de Vries Equation in a Quarter-Plane, Trans. Amer. Math. Soc., 2002, vol. 354, no. 2, pp. 427–490.

    Article  MATH  MathSciNet  Google Scholar 

  20. Volevich, L.R. and Gindikin, S.G., A Mixed Problem for (2b+1)-Hyperbolic Equations, Tr. Mosk. Mat. Obs., 1981, vol. 43, pp. 197–259.

    MathSciNet  Google Scholar 

  21. Yosida, K., Functional Analysis, Berlin: Springer, 1965. Translated under the title Funktsional’nyi analiz, Moscow: Mir, 1967.

    MATH  Google Scholar 

  22. Tartar, L., Interpolation non linéaire et régularité, J. Funct. Anal., 1972, vol. 9, pp. 469–489.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Peoples’ Friendship University of Russia, Moscow, Russia

    R. V. Kuvshinov & A. V. Faminskii

Authors
  1. R. V. Kuvshinov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Faminskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © R.V. Kuvshinov, A.V. Faminskii, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 3, pp. 391–402.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuvshinov, R.V., Faminskii, A.V. Mixed problem for the Kawahara equation in a half-strip. Diff Equat 45, 404–415 (2009). https://doi.org/10.1134/S0012266109030100

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266109030100

Keywords

Search

Navigation