Skip to main content
SpringerLink
Log in

Unique Continuation Property for the Benjamin Equation

  • Conference paper
  • First Online:
From Particle Systems to Partial Differential Equations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 75))

  • 1147 Accesses

Abstract

We show that the complex analysis techniques introduced in Bourgain (IMRN Int Math Res Not 9:437–447, 1997) can be used to prove that, if a sufficiently smooth solution to the Benjamin equation is compactly supported in a non trivial time interval then it vanishes identically.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Albert, J.P., Bona, J.L., Restrepo, J.M.: Solitary-wave solutions of the Benjamin equation. SIAM J. Appl. Math. 59, 2139–2161 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alvarez Samaniego, B., Angulo Pava, J.: Existence and stability of periodic travelling-wave solutions of the Benjamin equation. Commun. Pure Appl. Anal. 4, 367–388 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Angulo, J.: Existence and stability of solitary wave solutions of the Benjamin equation. J. Differ. Equ. 152, 136–159 (1999)

    Article  MATH  Google Scholar 

  4. Benjamin, T.B.: A new kind of solitary waves. J. Fluid Mech. 245, 401–411 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Benjamin, T.B.: Solitary and periodic waves of a new kind. Philos. Trans. R. Soc. Lond. Ser. A 354, 1775–1806 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourgain, J.: On the compactness of the support of solutions of dispersive equations. IMRN, Int. Math. Res. Not. 9, 437–447 (1997)

    Google Scholar 

  7. Bustamante, E., Isaza, P., Mejía, J.: On the support of solutions to the Zakharov-Kuznetsov equation. J. Funct. Anal. 264, 2529–2549 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carleman, T.: Sur les systèmes aux dérivées partielles du premier ordre à deux variables. C. R. Acad. Sci. Paris 97, 471–474 (1939)

    Google Scholar 

  9. Carvajal, X., Panthee, M.: Unique continuation property for a higher order nonlinear Schrödinger equation. J. Math. Anal. Appl. 303, 188–207 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carvajal, X., Panthee, M.: On uniqueness of solution for a nonlinear Schrödinger-Airy equation. Nonlinear Anal. 64(1), 146–158 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Escauriaza, L., Kenig, C.E., Ponce, G., Vega, L.: On uniqueness properties of solutions of Schrödinger equations. Commun. Partial Differ. Equ. 31, 1811–1823 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Escauriaza, L., Kenig, C.E., Ponce, G., Vega, L.: On uniqueness properties of solutions of the k-generalized KdV equations. J. Funct. Anal. 244, 504–535 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Esfahani, A., Pastor, A.: On the unique continuation property for Kadomtsev-Petviashvili-I and Benjamin-Ono-Zakharov-Kuznetsov equations. Bull. London Math. Soc. 43, 1130–1140 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fonseca, G., Ponce, G.: The IVP for the Benjamin-Ono equation in weighted Sobolev spaces. J. Funct. Anal. 260(2), 436–459 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fonseca, G., Linares, F., Ponce, G.: The IVP for the Benjamin-Ono equation in weighted Sobolev spaces II. J. Funct. Anal. 262, 2031–2049 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Commun. Math. Phys. 271(2), 511–522 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hormander, L.: Linear Partial Differential Operators. Springer, Berlin/Heidelberg/New York (1969)

    Book  Google Scholar 

  18. Iório, R.J., Jr.: Unique continuation principles for some equations of Benzamin-Ono Type. In: Nonlinear Equations: Methods, Models and Applications. Progr. Nonlinear Differential Equations Appl. vol. 54, pp. 163–179. Birkhäuser, Basel (2003)

    Google Scholar 

  19. Isakov, V.: Carleman type estimates in an anisotropic case and applications. J. Differ. Equ. 105, 217–238 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Isaza, P., Mejía, J.: On the support of solutions to the Ostrovsky equation with negative dispersion. J. Differ. Equ. 247(6), 1851–1865 (2009)

    Article  MATH  Google Scholar 

  21. Isaza, P., Mejía, J.: On the support of solutions to the Ostrovsky equation with positive dispersion. Nonlinear Anal. 72(11), 4016–4029 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Isaza, P., Mejía, J.: On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation. Commun. Pure Appl. Anal. 10, 1239–1255 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jiménez Urrea, J.: The Cauchy problem associated to the Benjamin equation in weighted Sobolev spaces. J. Differ. Equ. 254, 1863–1892 (2013)

    Article  MATH  Google Scholar 

  24. Kenig, C.E., Ponce, G., Vega, L.: On the support of solutions to the generalized KdV equation. Ann. Inst. Henri Poincare 19(2), 191–208 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kenig, C.E., Ponce, G., Vega, L.: On unique continuation for nonlinear Schrödinger equation. Commun. Pure Appl. Math. 56, 1247–1262 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kenig, C.E., Ponce, G., Vega, L.: On the unique continuation of solutions to the generalized KdV equation. Math. Res. Lett. 10, 833–846 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kozono, H., Ogawa, T., Tanisaka, H.: Well-posedness for the Benjamin equations. J. Korean Math. Soc. 38, 1205–1234 (2001)

    MATH  MathSciNet  Google Scholar 

  28. Linares, F.: L 2 global well-posedness of the initial value problem associated to the Benjamin equation. J. Differ. Equ. 152, 377–393 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  29. Mizohata, S.: Unicité du prolongement des solutions pour quelques operateurs differentials paraboliques. Mem. Coll. Sci. Univ. Kyoto A 31, 219–239 (1958)

    MATH  MathSciNet  Google Scholar 

  30. Panthee, M.: A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal. TMA 59, 425–438 (2004)

    MATH  MathSciNet  Google Scholar 

  31. Panthee, M.: Unique continuation property for the Kadomtsev-Petviashvili (KP-II) equation. Electron. J. Differ. Equ. 2005, 1–12 (2005)

    MathSciNet  Google Scholar 

  32. Panthee, M.: On the compact support of solutions to a nonlinear long internal waves model. Nepali Math. Sci. Rep. 24, 4958 (2005)

    MathSciNet  Google Scholar 

  33. Saut, J-C., Scheurer, B.: Unique continuation for some evolution equations. J. Differ. Equ. 66, 118–139 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  34. Tataru, D.: Carleman type estimates and unique continuation for the Schrödinger equation. Differ. Integral Equ. 8(4), 901–905 (1995)

    MATH  MathSciNet  Google Scholar 

  35. Zhang, B.Y.: Unique continuation for the Korteweg-de Vries equation. SIAM J. Math. Anal. 23, 55–71 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhou, Y.: Infinite propogation speed for a shallow water equation, preprint. http://www.fim.math.ethz.ch/preprints/2005/zhou.pdf

Download references

Acknowledgements

This work was partially supported by the Research Center of Mathematics of the University of Minho, Portugal through the FCT Pluriannual Funding Program, and FAPESP Brazil.

Author information

Authors and Affiliations

  1. CMAT-UMinho, 4710-057, Braga, Portugal

    Mahendra Panthee

  2. IMECC-UNICAMP, 13083-859, Campinas, SP, Brazil

    Mahendra Panthee

Authors
  1. Mahendra Panthee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahendra Panthee .

Editor information

Editors and Affiliations

  1. UMR CNRS-UNS 7351, Université de Nice Sophia Antipolis, Laboratoire J.A.Dieudonné, Nice, France

    Cédric Bernardin

  2. Universidade do Minho, Centre of Mathematics, Braga, Portugal

    Patricia Gonçalves

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Panthee, M. (2014). Unique Continuation Property for the Benjamin Equation. In: Bernardin, C., Gonçalves, P. (eds) From Particle Systems to Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54271-8_11

Download citation

Publish with us

Policies and ethics

Search

Navigation