Skip to main content
SpringerLink
Log in

Odd-Order Quasilinear Evolution Equations with General Nonlinearity on Bounded Intervals

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

An initial-boundary value problem posed on a bounded interval is considered for a class of odd-order (more than one) quasilinear evolution equations with general nonlinearity. Assumptions on the equations do not provide global a priori estimates for solutions of an arbitrary size. For small initial and boundary data, small right-hand side function results on global existence and uniqueness of small weak solutions, as well as on their large-time exponential decay are established.

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. F. D. Araruna, R. A. Capistrano-Filho, and G. G. Doronin, ‘‘Energy decay for the modified Kawahara equation posed in a bounded domain,’’ J. Math. Anal. Appl. 385, 743–756 (2012).

    Article  MathSciNet  Google Scholar 

  2. D. J. Benney, ‘‘Long waves on liquid films,’’ Stud. Appl. Math. 45, 150–155 (1966).

    MathSciNet  MATH  Google Scholar 

  3. D. J. Benney, ‘‘A general theory for interactions between short and long waves,’’ Stud. Appl. Math. 56, 81–94 (1977).

    Article  MathSciNet  Google Scholar 

  4. N. G. Berloff and L. N. Howard, ‘‘Solitary and periodic solutions for nonlinear nonintegrable equations,’’ Stud. Appl. Math. 99, 1–24 (1997).

    Article  MathSciNet  Google Scholar 

  5. J. P. Boyd, ‘‘Weakly non-local solitons for capillary-gravity waves: Fifth degree Korteweg–de Vries equation,’’ Phys. D (Amsterdam, Neth.) 48, 129–146 (1991).

  6. J. L. Bona, S.-M. Sun, and B.-Y. Zhang, ‘‘Nonhomogeneous problem for the Korteweg–de Vries equation in a bounded domain II,’’ J. Differ. Equat. 247, 2558–2596 (2009).

    Article  Google Scholar 

  7. T. J. Bridges and G. Derks, ‘‘Linear instability of solitary wave solutions of the Kawahara equation and its generalizations,’’ SIAM J. Math. Anal. 33, 1356–1378 (2002).

    Article  MathSciNet  Google Scholar 

  8. M. C. Caicedo, R. A. Capistrano-Filho, and B.-Y. Zang, ‘‘Control of the Korteweg–de Vries equation with Neumann boundary conditions,’’ SIAM J. Control Optim. 55, 3503–3532 (2017).

    Article  MathSciNet  Google Scholar 

  9. R. A. Capistrano-Filho, S.-M. Sun, and B.-Y. Zhang, ‘‘General boundary value problems of the Korteweg–de Vries equation on a bounded domain,’’ Math. Control Relat. Fields 8, 583–605 (2018).

    Article  MathSciNet  Google Scholar 

  10. J. Ceballos, M. Sepulveda, and O. Villagran, ‘‘The Korteweg–de Vries–Kawahara equation in a bounded domain and some numerical results,’’ Appl. Math. Comput. 190, 912–936 (2007).

    MathSciNet  MATH  Google Scholar 

  11. A. V. Faminskii, ‘‘The Cauchy problem for quasilinear equations of odd order,’’ Math. USSR-Sb. 68, 31–59 (1991).

    Article  MathSciNet  Google Scholar 

  12. A. V. Faminskii and N. A. Larkin, ‘‘Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval,’’ Electron. J. Differ. Equat. 2010 (1), 1–20 (2010).

    MathSciNet  MATH  Google Scholar 

  13. A. V. Faminskii and N. A. Larkin, ‘‘Odd-order evolution equations posed on a bounded interval,’’ Bol. Soc. Paranaense Mat. 28, 67–77 (2010).

    MathSciNet  MATH  Google Scholar 

  14. O. Glass and S. Guerrero, ‘‘On the controllability of the fifth order Korteweg–de Vries equation,’’ Ann. Inst. H. Poincare Non-Lin. Anal. 26, 2181–2209 (2009).

    MATH  Google Scholar 

  15. J. K. Hunter and J. Scheurle, ‘‘Existence of perturbed solitary wave solutions to a model equation for water waves,’’ Phys. D (Amsterdam, Neth.) 32, 253–268 (1988).

  16. J. Holmer, ‘‘The initial boundary value problem for the Korteweg–de Vries equation,’’ Comm. Partial Differ. Equat. 31, 1151–1190 (2006).

    Article  MathSciNet  Google Scholar 

  17. Y. Jia and Z. Huo, ‘‘Well-posedness for the fifth-order shallow water equations,’’ J. Differ. Equat. 246, 2448–2467 (2009).

    Article  MathSciNet  Google Scholar 

  18. R. S. Johnston, ‘‘A nonlinear equation incorporating damping and dispersion,’’ J. Fluid Mech. 42, 49–60 (1970).

    Article  MathSciNet  Google Scholar 

  19. P. Isaza, F. Linares, and G. Ponce, ‘‘Decay properties for solutions of fifth order nonlinear dispersive equations,’’ J. Differ. Equat. 258, 764–795 (2015).

    Article  MathSciNet  Google Scholar 

  20. D. J. Kaup, ‘‘On the inverse scattering problem for cubic eigenvalue problems of the class \(\psi_{xxx}+6Q\psi_{x}+6R\psi=\lambda\psi\),’’ Stud. Appl. Math. 62, 189–216 (1980).

    Article  MathSciNet  Google Scholar 

  21. T. Kawahara, ‘‘Oscillatory solitary waves in dispersive media,’’ J. Phys. Soc. Jpn. 33, 260–264 (1972).

    Article  Google Scholar 

  22. C. E. Kenig and D. Pilod, ‘‘Local well-posedness for the KdV hierarchy at high regularity,’’ Adv. Differ. Equat. 21, 801–836 (2015).

    MathSciNet  MATH  Google Scholar 

  23. C. Kenig, G. Ponce, and L. Vega, ‘‘Higher order nonlinear dispersive equations,’’ Proc. Am. Math. Soc. 122, 157–166 (1994).

    Article  MathSciNet  Google Scholar 

  24. N. Khanal, J. Wu, and J.-M. Yuan, ‘‘The Kawahara equation in weighted Sobolev spaces,’’ Nonlinearity 21, 1489–1505 (2008).

    Article  MathSciNet  Google Scholar 

  25. S. Kichenassamy and P. J. Olver, ‘‘Existence and nonexistence of solitary wave solutions to higher-order model evolution equations,’’ SIAM J. Math. Anal. 23, 1141–1166 (1992).

    Article  MathSciNet  Google Scholar 

  26. D. J. Korteweg and G. de Vries, ‘‘On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,’’ Philos. Mag. 39, 422–443 (1895).

    Article  MathSciNet  Google Scholar 

  27. E. F. Kramer and B.-Y. Zhang, ‘‘Nonhomogeneous boundary value problems for the Korteweg–de Vries equation on a bounded domain,’’ J. Syst. Sci. Complex. 23, 499-526 (2010).

    Article  MathSciNet  Google Scholar 

  28. E. F. Kramer, I. Rivas, and B.-Y. Zhang, ‘‘Well-posedness of a class of non-homogeneous boundary value problem of the Korteweg–de Vries equation on a finite domain,’’ ESAIM: Control Optim. Calc. Var. 19, 358–384 (2013).

    MathSciNet  MATH  Google Scholar 

  29. B. A. Kupershmidt, ‘‘A super Korteweg–de Vries equation: An integrable system,’’ Phys. Lett. A 102, 213–215 (1984).

    Article  MathSciNet  Google Scholar 

  30. S. Kwon, ‘‘Well-posedness and ill-posedness of the fifth order KdV equation,’’ Electron. J. Differ. Equat. 2008, 1–15 (2008).

    MathSciNet  MATH  Google Scholar 

  31. N. A. Larkin and J. Luchesi, ‘‘General mixed problems for the KdV equations on bounded intervals,’’ Electron. J. Differ. Equat. 2010 (168), 1–17 (2010).

    MathSciNet  MATH  Google Scholar 

  32. N. A. Larkin and J. Luchesi, ‘‘Initial-boundary value problems for generalized dispersive equations of higher orders posed on boundary intervals,’’ J. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09579-w

  33. N. A. Larkin and M. H. Simões, ‘‘General boundary conditions for the Kawahara equation on bounded intervals,’’ Electron. J. Differ. Equat. 2013 (159), 1–21 (2013).

    Article  MathSciNet  Google Scholar 

  34. N. A. Larkin and M. H. Simões, ‘‘The Kawahara equation on bounded intervals and on a half-line,’’ Nonlin. Anal. 127, 397–412 (2015).

    Article  MathSciNet  Google Scholar 

  35. P. D. Lax, ‘‘Integrals of nonlinear equations of evolution and solitary waves,’’ Commun. Pure Appl. Math. 21, 467–490 (1965).

    Article  MathSciNet  Google Scholar 

  36. S. P. Lin, ‘‘Finite amplitude side-band stability of a viscous film,’’ J. Fluid Mech. 63, 417–429 (1974).

    Article  Google Scholar 

  37. E. Lisher, ‘‘Comments on the use of the Korteweg–de Vries equation in the study of anharmonic lattices,’’ Proc. R. Soc. London, Ser. A 339, 119–126 (1974).

    Article  Google Scholar 

  38. G. Ponce, ‘‘Lax pairs and higher order models for water waves,’’ J. Differ. Equat. 102, 360–381 (1993).

    Article  MathSciNet  Google Scholar 

  39. I. Rivas, M. Usman, and B.-Y. Zhang, ‘‘Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg–de Vries equation on a finite domain,’’ Math. Control Relat. Fields 1, 61–81 (2011).

    Article  MathSciNet  Google Scholar 

  40. J.-C. Saut, ‘‘Quelques généralisations de l’équation de Korteweg–de Vries. II,’’ J. Differ. Equat. 33, 320–335 (1979).

    Article  Google Scholar 

  41. S. P. Tao and S. B. Cui, ‘‘Local and global existence to initial value problems of nonlinear Kaup–Kupershmidt equations,’’ Acta Mat. Sinica, Engl. Ser. 21, 881–892 (2005).

    MATH  Google Scholar 

  42. J. Topper and T. Kawahara, ‘‘Approximate equations for long nonlinear waves on a viscous fluid,’’ J. Phys. Soc. Jpn. 44, 663–666 (1978).

    Article  MathSciNet  Google Scholar 

Download references

Funding

The work was supported by the Ministry of Science and Higher Education of Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018).

Author information

Authors and Affiliations

  1. Peoples’ Friendship University of Russia (RUDN University), 117198, Moscow, Russia

    A. V. Faminskii

Authors
  1. A. V. Faminskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Faminskii.

Additional information

(Submitted by A. B. Muravnik)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faminskii, A.V. Odd-Order Quasilinear Evolution Equations with General Nonlinearity on Bounded Intervals. Lobachevskii J Math 42, 875–888 (2021). https://doi.org/10.1134/S1995080221050048

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080221050048

Keywords:

Search

Navigation