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Advances in Mathematical Physics
Volume 2015, Article ID 269536, 13 pages
http://dx.doi.org/10.1155/2015/269536
Research Article

Kawahara-Burgers Equation on a Strip

Departamento de Matemática, Universidade Estadual de Maringá, 87020-900 Maringá, PR, Brazil

Received 13 August 2015; Accepted 11 November 2015

Academic Editor: Mikhail Panfilov

Copyright © 2015 N. A. Larkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C. J. Amick, J. L. Bona, and M. E. Schonbek, “Decay of solutions of some nonlinear wave equations,” Journal of Differential Equations, vol. 81, no. 1, pp. 1–49, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. D. J. Benney, “A general theory for interactions between short and long waves,” Studies in Applied Mathematics, vol. 56, no. 1, pp. 81–94, 1977. View at Google Scholar
  3. H. A. Biagioni and F. Linares, “On the Benney-Lin and Kawahara equations,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 131–152, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. J. L. Bona, S. M. Sun, and B.-Y. Zhang, “Nonhomogeneous problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane,” Annales de l'Institut Henri Poincaré—analyse Non Linéaire, vol. 25, pp. 1145–1185, 2008. View at Google Scholar
  5. W. Chen and Z. Guo, “Globalwell-posedness and I method for the fifth order Korteweg-de Vries equation,” Journal d'Analyse Mathématique, vol. 114, no. 1, pp. 121–156, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. Ceballos, M. Sepulveda, and O. Villagran, “The Korteweg-de Vries-Kawahara equation in a bounded domain and some numerical results,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 912–936, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. S. B. Cui, D. G. Deng, and S. P. Tao, “Global existence of solutions for the Cauchy problem of the Kawahara equation with L2 initial data,” Acta Mathematica Sinica, vol. 22, no. 5, pp. 1457–1466, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. P. Isaza, F. Linares, and G. Ponce, “Decay properties for solutions of fifth order nonlinear dispersive equations,” Journal of Differential Equations, vol. 258, no. 3, pp. 764–795, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. S. Kwon, “On the fifth-order KdV equation: local well-posedness and lack of uniform continuity of the solution map,” Journal of Differential Equations, vol. 245, no. 9, pp. 2627–2659, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. T. Kawahara, “Oscillatory solitary waves in dispersive media,” Journal of the Physical Society of Japan, vol. 33, no. 1, pp. 260–264, 1972. View at Publisher · View at Google Scholar · View at Scopus
  11. V. E. Zakharov and E. A. Kuznetsov, “On three-dimensional solitons,” Soviet Physics—JETP, vol. 39, pp. 285–286, 1974. View at Google Scholar
  12. T. Kato, “On the Cauchy problem for the (generalized) Korteweg-de Vries equations,” in Advances in Mathematics Suplementary Studies, vol. 8 of Studies in Applied Mathematics, pp. 93–128, Academic Press, New York, NY, USA, 1983. View at Google Scholar
  13. T. Kato, “Local well-posedness for Kawahara equation,” Advances in Differential Equations, vol. 16, no. 3-4, pp. 257–287, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  14. C. E. Kenig, G. Ponce, and L. Vega, “Higher-order nonlinear dispersive equations,” Proceedings of the American Mathematical Society, vol. 122, no. 1, pp. 157–166, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,” Communications on Pure and Applied Mathematics, vol. 46, no. 4, pp. 527–620, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Ponce, “Regularity of solutions to nonlinear dispersive equations,” Journal of Differential Equations, vol. 78, no. 1, pp. 122–135, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J.-C. Saut, “Sur quelques généralizations de l'équation de Korteweg- de Vries,” Journal de Mathématiques Pures et Appliquées, vol. 58, no. 1, pp. 21–61, 1979. View at Google Scholar · View at MathSciNet
  18. G. G. Doronin and N. A. Larkin, “Boundary value problems for the stationary Kawahara equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 5-6, pp. 1655–1665, 2008. View at Publisher · View at Google Scholar · View at Scopus
  19. G. G. Doronin and N. A. Larkin, “Kawahara equation in a bounded domain,” Discrete and Continuous Dynamical Systems, Series B, vol. 10, no. 4, pp. 783–799, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  20. R. V. Kuvshinov and A. V. Faminskii, “A mixed problem in a half-strip for the Kawahara equation,” Differentsial'nye Uravneniya, vol. 45, no. 3, pp. 391–402, 2009 (Russian), Translation in Differential Equations, vol. 45 no. 3, pp. 404–415, 2009. View at Google Scholar
  21. N. A. Larkin, “Correct initial boundary value problems for dispersive equations,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 1079–1092, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. N. A. Larkin and G. G. Doronin, “Kawahara equation in a quarter-plane and in a finite domain,” Boletim da Sociedade Paranaense de Matemática (3s), vol. 25, no. 1-2, pp. 9–16, 2007. View at Google Scholar
  23. N. A. Larkin, “Korteweg-de Vries and Kuramoto-Sivashinsky equations in bounded domains,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 169–185, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. N. A. Larkin, “The 2D Kawahara equation on a half-strip,” Applied Mathematics & Optimization, vol. 70, no. 3, pp. 443–468, 2014. View at Publisher · View at Google Scholar
  25. N. A. Larkin, “Exponential decay of the H1-norm for the 2D Zakharov-Kuznetsov equation on a half-strip,” Journal of Mathematical Analysis and Applications, vol. 405, no. 1, pp. 326–335, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. N. A. Larkin and M. H. Simoes, “General boundary conditions for the Kawahara equation on bounded intervals,” Electronic Journal of Differential Equations, vol. 159, pp. 1–21, 2013. View at Google Scholar
  27. A. V. Faminskii and N. A. Larkin, “Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval,” Electronic Journal of Differential Equations, vol. 2010, no. 1, pp. 1–20, 2010. View at Google Scholar · View at MathSciNet
  28. N. A. Larkin, “The 2D Zakharov-Kuznetsov-Burgers equation on a strip,” Boletim da Sociedade Paranaense de Matemática, vol. 34, no. 1, pp. 151–172, 2016. View at Publisher · View at Google Scholar
  29. S. A. Elwakil, E. K. El-Shewy, and H. G. Abdelwahed, “Solution of the perturbed Zakharov-Kuznetsov (ZK) equation describing electron-acoustic solitary waves in a magnetized plasma,” Chinese Journal of Physics, vol. 49, no. 3, pp. 732–744, 2011. View at Google Scholar · View at Scopus
  30. A. V. Faminskii, “The Cauchy problem for the Zakharov-Kuznetsov equation,” Differentsial'nye Uravneniya, vol. 31, pp. 1070–1081, 1995 (Russian), English Translation in: Differential Equations, vol. 31, pp. 1002–1012, 1995. View at Google Scholar
  31. L. G. Farah, F. Linares, and A. Pastor, “A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results,” Journal of Differential Equations, vol. 253, no. 8, pp. 2558–2571, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. F. Linares and A. Pastor, “Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation,” SIAM Journal on Mathematical Analysis, vol. 41, no. 4, pp. 1323–1339, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. F. Linares and J.-C. Saut, “The Cauchy problem for the 3D Zakharov-Kuznetsov equation,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 2, pp. 547–565, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. F. Ribaud and S. Vento, “Well-posedness results for the three-dimensional Zakharov-Kuznetsov equation,” SIAM Journal on Mathematical Analysis, vol. 44, no. 4, pp. 2289–2304, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. E. S. Baykova and A. V. Faminskii, “On initial-boundary value problems in a strip for the generalized two-dimensional Zakharov-Kuznetsov equation,” Advances in Differential Equations, vol. 18, no. 7-8, pp. 663–686, 2013. View at Google Scholar · View at MathSciNet · View at Scopus
  36. A. V. Faminskii, “Weak solutions to initial-boundary value problems for quasilinear evolution equations of any odd order,” Advances in Differential Equations, vol. 17, no. 5-6, pp. 421–470, 2012. View at Google Scholar
  37. A. V. Faminskii, “An initial-boundary value problem in a strip for a two-dimensional equation of Zakharov-Kuznetsov type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 116, no. 4, pp. 132–144, 2015. View at Publisher · View at Google Scholar
  38. N. A. Larkin and E. Tronco, “Regular solutions of the 2D Zakharov-Kuznetsov equation on a half-strip,” Journal of Differential Equations, vol. 254, no. 1, pp. 81–101, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. J.-C. Saut and R. Temam, “An initial boundary-value problem for the Zakharov-Kuznetsov equation,” Advances in Differential Equations, vol. 15, no. 11-12, pp. 1001–1031, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  40. G. G. Doronin and N. A. Larkin, “Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip,” Proceedings of the Edinburgh Mathematical Society, vol. 58, no. 3, pp. 661–682, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  41. L. Rosier and B.-Y. Zhang, “Control and stabilization of the Korteweg-de Vries equation: recent progresses,” Journal of Systems Science and Complexity, vol. 22, no. 4, pp. 647–682, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  42. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, USA, 1968.