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Initial-Boundary Value Problems for Generalized Dispersive Equations of Higher Orders Posed on Bounded Intervals

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Abstract

Initial-boundary value problems for nonlinear dispersive equations of evolution of order \(2l+1, \;l\in \mathbb {N}\) with a convective term of the form \(u^ku_x,\;k\in \mathbb {N}\) have been considered on intervals \((0,L),\;L\in (0,+\infty )\). Definitions of regular and critical values of k as functions of l have been given. The existence and uniqueness of global regular solutions as well as exponential decay of them for small initial data have been established.

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References

  1. Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Elsevier Science Ltd, Amsterdam (2003)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Biagioni, H.A., Linares, F.: On the Benney—Lin and Kawahara equations. J. Math. Anal. Appl. 211, 131–152 (1997)

    Article  MathSciNet  Google Scholar 

  4. Ceballos, J., Sepulveda, M., Villagran, O.: 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 

  5. Doronin, G.G., Larkin, N.A.: Kawahara equation in a bounded domain. Discret. Contin. Dyn. Syst. Ser. B 10, 783–799 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Faminskii, A.V.: Cauchy problem for quasilinear equation of odd order. Mat. Sb. 180, 1183–1210 (1989). Transl. in Math. USSR-Sb., 68 (1991), 31-59. 1979

    Google Scholar 

  7. Faminskii, A.V., Larkin, N.A.: Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval. Electron. J. Differ. Equ. 1–20 (2010)

  8. Farah, L.G., Linares, F., Pastor, A.: The supercritical generalized KDV equation: global well-posedness in the energy space and below. Math. Res. Lett. 18(02), 357–377 (2011)

    Article  MathSciNet  Google Scholar 

  9. Fonseca, G., Linares, F., Ponce, G.: Global well-posedness for the modified Korteweg-de Vries equation. Commun. Partial Differ. Equ. 24(3,4), 683–705 (1999)

    Article  MathSciNet  Google Scholar 

  10. Fonseca, G., Linares, F., Ponce, G.: Global existence for the critical generalized KDV equation. Proc. AMS 131(6), 1847–1855 (2002)

    Article  MathSciNet  Google Scholar 

  11. Huo, Z., Jia, Y.: Well-posedness for the fifth-order shallow water equations. J. Differ. Equ. 246, 2448–2467 (2009)

    Article  MathSciNet  Google Scholar 

  12. Isaza, P., Linares, F., Ponce, G.: Decay properties for solutions of fifth order nonlinear dispersive equations. J. Differ. Equ. 258, 764–795 (2015)

    Article  MathSciNet  Google Scholar 

  13. Jeffrey, A., Kakutani, T.: Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation. SIAM Rev. 14(4), 582–643 (1972)

    Article  MathSciNet  Google Scholar 

  14. Kakutani, T., Ono, H.: Weak non linear hydromagnetic waves in a cold collision free plasma. J. Phys. Soc. Jpn. 26, 1305–1318 (1969)

    Article  Google Scholar 

  15. Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Stud. Appl. Math. Adv. Math. Suppl. Stud. 8, 93–128 (1983)

    MathSciNet  Google Scholar 

  16. Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33, 260–264 (1972)

    Article  Google Scholar 

  17. Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation and the contraction principle. Commun. Pure Appl. Math. 46(4), 527–620 (1993)

    Article  MathSciNet  Google Scholar 

  18. Kenig, C.E., Ponce, G., Vega, L.: Higher-order nonlinear dispersive equations. Proc. Am. Math. Soc. 122(1), 157–166 (1994)

    Article  MathSciNet  Google Scholar 

  19. Kuvshinov, R.V., Faminskii, A.V.: A mixed problem in a half-strip for the Kawahara equation (Russian). Differ. Uravn. 45(3), 391–402 (2009). translation in Differ. Equ. 45 N. 3, 404–415 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Larkin, N.A.: Korteweg-de Vries and Kuramoto-Sivashinsky equations in bounded domains. J. Math. Anal. Appl. 297, 169–185 (2004)

    Article  MathSciNet  Google Scholar 

  21. Larkin, N.A.: Correct initial boundary value problems for dispersive equations. J. Math. Anal. Appl. 344, 1079–1092 (2008)

    Article  MathSciNet  Google Scholar 

  22. Larkin, N.A., Luchesi, J.: On connection between the order of a stationary one-dimensional dispersive equation and the growth of its convective term. Bol. Soc. Paran. Mat. (2018) (in press) https://doi.org/10.5269/bspm.41478

  23. Larkin, N.A., Luchesi, J.: Generalized dispersive equations of higher orders posed on bounded intervals: local theory. arXiv:1812.04146v1 [math.AP]. Accessed 10 Dec 2018

  24. Larkin, N.A., Simões, M.H.: The Kawahara equation on bounded intervals and on a half-line. Nonlinear Anal. 127, 397–412 (2015)

    Article  MathSciNet  Google Scholar 

  25. Linares, F., Pazoto, A.: On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping. Proc. Am. Math. Soc. 135(1), 1515–1522 (2007)

    Article  MathSciNet  Google Scholar 

  26. Martel, Y., Merle, F.: Instability of solutions for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal. 11, 74–123 (2001)

    Article  MathSciNet  Google Scholar 

  27. Merle, F.: Existence of blow up solutions in the energy space for the critical generalized KdV equation. J. Am. Math. Soc. 14, 555–578 (2001)

    Article  MathSciNet  Google Scholar 

  28. Nazarov A.I., Kuznetsov N.G., Poborchi S.V.: V. A. Steklov and problem of sharp (exact) constants in inequalities of mathematical physics, arXiv:1307.8025v1 [math.OH]. Accessed 30 July 2013

  29. Nirenberg, L.: On elliptic partial differential equations, Annali della Scuola Nomale Superiore di Pisa, Classe di Scienze 3 srie tome 13(2), 115–162 (1959)

    Google Scholar 

  30. Pilod, D.: On the Cauchy problem for higher-order nonlinear dispersive equations. J. Differ. Equ. 245, 2055–2077 (2008)

    Article  MathSciNet  Google Scholar 

  31. Rosier, L., Zhang, B.-Y.: Global stabilization of the generalized Korteweg-de Vries equation on a finite domain. SIAM J. Control Optim. 45(3), 927–956 (2006)

    Article  MathSciNet  Google Scholar 

  32. Sangare, K., Faminskii, A.V.: Weak solutions of a mixed problem in a half-strip for a generalized Kawahara equation. Math. Notes 85, 90–100 (2009)

    Article  MathSciNet  Google Scholar 

  33. Saut, J.-C.: Sur quelques généralizations de l’équation de Korteweg- de Vries. J. Math. Pures Appl. 58, 21–61 (1979)

    MathSciNet  MATH  Google Scholar 

  34. Tao, S.P., Cui, S.B.: The local and global existence of the solution of the Cauchy problem for the seven-order nonlinear equation. Acta Math. Sinica 25 A(4), 451–460 (2005)

    Google Scholar 

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Acknowledgements

We appreciate profound and helpful comments of the reviewer. Nikolai A. Larkin has been supported by Fundação Araucária, Paraná, Brazil; convenio No. 307/2015, Protocolo No. 45.703.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo 5790: Agência UEM, Maringá, PR, 87020-900, Brazil

    N. A. Larkin

  2. Departamento de Matemática, Universidade Tecnológica Federal do Paraná - Câmpus Pato Branco, Via do Conhecimento Km 1, Pato Branco, PR, 85503-390, Brazil

    J. Luchesi

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Larkin, N.A., Luchesi, J. Initial-Boundary Value Problems for Generalized Dispersive Equations of Higher Orders Posed on Bounded Intervals. Appl Math Optim 83, 1081–1102 (2021). https://doi.org/10.1007/s00245-019-09579-w

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