Skip to main content
SpringerLink
Log in

Global Time Decay of the Amplitude of a Reflected Wave

  • Conference paper
Partial Differential Equations and Mathematical Physics

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 21))

  • 661 Accesses

Abstract

We are interested in global time estimates for the L P norms of solutions to the wave equation on the exterior of a smooth compact strictly convex obstacle in R n n 2265 2, with vanishing Dirichlet data on the boundary:

$$\square u = 0\,on\,\Omega \,x\,R;u(0) = {f_0},\,{u_t}(0) = {f_1},{\left. u \right|_{\partial \Omega xR}} = 0.$$
(1.1)

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Beals, D’ boundedness of Fourier integral operators, Mem. Amer. Math. Soc. 38 (No 264) (1982).

    Google Scholar 

  2. M. Beals, Local L“ estimates for Fourier-Airy integral operators, Comm. Pure Appl. Math. 35 (1982), 751–769.

    MathSciNet  MATH  Google Scholar 

  3. M. Beals, Optimal L°O decay for solutions to the wave equation with a potential, Comm. in PDE 19 (1994), 1319–1369.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Beals and W. Strauss, Time decay estimates for a perturbed wave equation, Journées Equations aux dérivées partielles, St Jean de Monts (1992), Exp. no XIII.

    Google Scholar 

  5. M. Beals and W. Strauss, LP estimates for the wave equation with a potential, Comm. in PDE 18 (1993), 1365–1397.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z. 189 (1985), 487–505.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27–42.

    MATH  Google Scholar 

  9. L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79–183.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. D. Lax and R. S. Phillips, Scattering Theory, Academic Press, New York, London, 1967.

    Google Scholar 

  11. H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations,J. Funct. Anal. (to appear).

    Google Scholar 

  12. B. Marshall, W. Strauss and S. Wainger, LP -.L 9 estimates for the Klein-Gordon equations, J. Math. Pures Appl. (A) 59 (1980), 417–440.

    MathSciNet  MATH  Google Scholar 

  13. R. B. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J. 46 (1979), 43–59.

    MathSciNet  MATH  Google Scholar 

  14. R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems I, Comm. Pure Appl. Math. 31 (1978), 593–617.

    MATH  Google Scholar 

  15. C. S. Morawetz, J. V. Ralston and W. A. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles,Comm. Pure Appl. Math. 30 (1977), 447508.

    Google Scholar 

  16. J. Peral, LP estimates for the wave equation, J. Funct. Anal. 36 (1980), 114–145.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Ralston, Note on the decay of acoustic waves, Duke Math. J. 46 (1979), 799–804.

    MathSciNet  MATH  Google Scholar 

  18. J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. 138 (1993), 503–518.

    Article  MathSciNet  MATH  Google Scholar 

  19. H. F. Smith and C. D. Sogge, LP regularity for the wave equation with strictly convex obstacles, Duke Math. J 73 (1994), 97–155.

    MathSciNet  MATH  Google Scholar 

  20. H. F. Smith and C. D. Sogge, On the critical semilinear wave equation outside convex obstacles, Preprint.

    Google Scholar 

  21. W. A. Strauss, Nonlinear scattering at low energy, J. Funct. Anal. 41 (1981), 110133.

    Google Scholar 

  22. W. A. Strauss, Nonlinear Wave Equations, CBMS Reg. Conf. Series in Math. no 73, Amer. Math. Soc., Providence (1989).

    Google Scholar 

  23. R. S. Strichartz, A priori estimates for the wave equation and some applications, J. Funct. Anal. 5 (1970), 218–235.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, 1981.

    MATH  Google Scholar 

  25. B. R. Vainberg, On the short wave asymptotic behavior of solutions of stationary problems., Russ. Math. Surveys 30:2 (1975), 1–58.

    Google Scholar 

  26. K. Yajima, The LP continuity of wave operators for Schrödinger operators,Preprint.

    Google Scholar 

  27. Jiaping Zhong, The Lp-L 9 estimates for the wave equation with a nonnegative potential,Comm. in PDE 20 (1995), 315–334.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    Michael Beals

Authors
  1. Michael Beals
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Lund, S-221 00, Lund, Sweden

    Lars Hörmander  & Anders Melin  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer Science+Business Media New York

About this paper

Cite this paper

Beals, M. (1996). Global Time Decay of the Amplitude of a Reflected Wave. In: Hörmander, L., Melin, A. (eds) Partial Differential Equations and Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 21. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0775-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0775-7_3

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6897-0

  • Online ISBN: 978-1-4612-0775-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation