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On the wave operator for the generalized Boussinesq equation


Authors: Luiz Gustavo Farah and Lucas C. F. Ferreira
Journal: Proc. Amer. Math. Soc. 140 (2012), 3055-3066
MSC (2010): Primary 35Q35; Secondary 35B40, 35P25
Published electronically: December 28, 2011
MathSciNet review: 2917079
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Abstract: We study the asymptotic behavior of solutions for the generalized Boussinesq equation in a singular framework. We construct a wave operator (inverse scattering) for large profiles $ \vec {h}$ belonging to an infinite mass framework based on weak $ L^{p}$-spaces. Our solutions converge towards the prescribed scattering state with a polynomial rate.


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  • 1. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • 2. Jerry L. Bona and Robert L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys. 118 (1988), no. 1, 15–29. MR 954673
  • 3. J. Boussinesq, Théorie des ondes et de remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide dans ce canal des vitesses sensiblement de la surface au fond, J. Math. Pures Appl. 17 (1872), 55-108.
  • 4. Yonggeun Cho and Tohru Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Discrete Contin. Dyn. Syst. 17 (2007), no. 4, 691–711. MR 2276469, 10.3934/dcds.2007.17.691
  • 5. Yonggeun Cho and Tohru Ozawa, Global existence on nonlinear Schrödinger-IMBq equations, J. Math. Kyoto Univ. 46 (2006), no. 3, 535–552. MR 2311358
  • 6. Raphaël Côte, Large data wave operator for the generalized Korteweg-de Vries equations, Differential Integral Equations 19 (2006), no. 2, 163–188. MR 2194502
  • 7. F. Falk, E. Laedke, and K. Spatschek, Stability of solitary-wave pulses in shape-memory alloys, Phys. Rev. B 36 (6) (1987), 3031-3041.
  • 8. Luiz Gustavo Farah, Large data asymptotic behaviour for the generalized Boussinesq equation, Nonlinearity 21 (2008), no. 2, 191–209. MR 2384545, 10.1088/0951-7715/21/2/001
  • 9. Luiz Gustavo Farah, Local solutions in Sobolev spaces with negative indices for the “good” Boussinesq equation, Comm. Partial Differential Equations 34 (2009), no. 1-3, 52–73. MR 2512853, 10.1080/03605300802682283
  • 10. Luiz Gustavo Farah, Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation, Commun. Pure Appl. Anal. 8 (2009), no. 5, 1521–1539. MR 2505284, 10.3934/cpaa.2009.8.1521
  • 11. J. Ginibre, T. Ozawa, and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 2, 211–239 (English, with English and French summaries). MR 1270296
  • 12. J. Ginibre and G. Velo, Long range scattering for some Schrödinger related nonlinear systems, preprint, arXiv:math/0412430v1, 2004.
  • 13. Felipe Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations 106 (1993), no. 2, 257–293. MR 1251854, 10.1006/jdeq.1993.1108
  • 14. Felipe Linares and Márcia Scialom, Asymptotic behavior of solutions of a generalized Boussinesq type equation, Nonlinear Anal. 25 (1995), no. 11, 1147–1158. MR 1350736, 10.1016/0362-546X(94)00236-B
  • 15. Yue Liu, Decay and scattering of small solutions of a generalized Boussinesq equation, J. Funct. Anal. 147 (1997), no. 1, 51–68. MR 1453176, 10.1006/jfan.1996.3052
  • 16. D. H. Peregrine, Equations for water waves and the approximations behind them, Waves on Beaches and Resulting Sediment Transport, Academic Press, New York, 1972, 95-122.
  • 17. Akihiro Shimomura, Scattering theory for the Schrödinger-improved Boussinesq system in two space dimensions, Asymptot. Anal. 51 (2007), no. 2, 167–187. MR 2311159
  • 18. Masayoshi Tsutsumi and Tomomi Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon. 36 (1991), no. 2, 371–379. MR 1095753
  • 19. Suxia Xia and Jia Yuan, Existence and scattering of small solutions to a Boussinesq type equation of sixth order, Nonlinear Anal. 73 (2010), no. 4, 1015–1027. MR 2653768, 10.1016/j.na.2010.04.028

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Additional Information

Luiz Gustavo Farah
Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
Email: lgfarah@gmail.com

Lucas C. F. Ferreira
Affiliation: Departamento de Matemática, Universidade Estadual de Campinas, CEP 13083-859, Campinas-SP, Brazil
Email: lcff@ime.unicamp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11131-6
Keywords: Boussinesq equation, inverse scattering, large data, Lorentz spaces.
Received by editor(s): October 1, 2010
Received by editor(s) in revised form: March 16, 2011
Published electronically: December 28, 2011
Additional Notes: The first author was partially supported by FAPEMIG and CNPq, Brazil.
The second author was supported by CNPq and FAPESP, Brazil
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.