Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle


Author: Jason L. Metcalfe
Journal: Trans. Amer. Math. Soc. 356 (2004), 4839-4855
MSC (2000): Primary 35L05
DOI: https://doi.org/10.1090/S0002-9947-04-03667-0
Published electronically: June 25, 2004
MathSciNet review: 2084401
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. In order to prove the global estimates, we explore weighted Strichartz estimates for solutions of the wave equation when the Cauchy data and forcing term are compactly supported.


References [Enhancements On Off] (What's this?)

  • 1. N. Burq: Global Strichartz estimates for nontrapping geometries: A remark about an article by H. Smith and C. Sogge, Comm. Partial Differential Equations, 28, (2003), no. 9-10, 1675-1683. MR 2001179 (2004g:35146)
  • 2. M. Christ, A. Kiselev: Maximal functions associated to filtrations, J. Funct. Anal., 179, (2001), 409-425. MR 1809116 (2001i:47054)
  • 3. L. Hörmander: Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, 1997. MR 1466700 (98e:35103)
  • 4. M. Keel, T. Tao: Endpoint Strichartz Estimates, Amer J. Math., 120, (1998), 955-980. MR 1646048 (2000d:35018)
  • 5. P. D. Lax, R. S. Philips: Scattering Theory (Revised Edition), Academic Press Inc., 1989. MR 1037774 (90k:35005)
  • 6. R. Melrose: Singularities and energy decay in acoustical scattering, Duke Math. J., 46, (1979), 43-59. MR 0523601 (80h:35104)
  • 7. C. Morawetz: Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28, (1975), 229-264. MR 0372432 (51:8641)
  • 8. C. Morawetz: The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14, (1961), 561-568. MR 0132908 (24:A2744)
  • 9. C. Morawetz: Exponential Decay of Solutions of the Wave Equation, Comm. Pure Appl. Math., 19, (1966), 439-444. MR 0204828 (34:4664)
  • 10. C. Morawetz, J. Ralston, W. Strauss: Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math., 30, (1977), 447-508. MR 0509770 (58:23091a)
  • 11. J. Ralston: Note on the decay of acoustic waves, Duke Math. J., 46, (1979), 799-804. MR 0552527 (80m:35051)
  • 12. H. Smith, C.D. Sogge: Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations, 25, (2000), 2171-2183. MR 1789924 (2001j:35180)
  • 13. H. Smith, C.D. Sogge: On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc., 8, (1995), 879-916. MR 1308407 (95m:35128)
  • 14. W. Strauss: Dispersal of waves vanishing on the boundary of an exterior domain, Comm. Pure Appl. Math., 28, (1975), 265-278. MR 0367461 (51:3703)
  • 15. R. Strichartz: A priori estimates for the wave equation and some applications, J. Funct. Analysis, 5, (1970), 218-235. MR 0257581 (41:2231)
  • 16. R. Strichartz: Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke Math. J., 44, (1977), 705-714. MR 0512086 (58:23577)
  • 17. M. Taylor: Grazing rays and reflection of singularities of solutions to wave equations, Comm. Pure Appl. Math., 29, (1976), 1-38. MR 0397175 (53:1035)
  • 18. M. Taylor: Partial differential equations I, Springer-Verlag, Berlin, 1996. MR 1395148 (98b:35002b)
  • 19. B. R. Vainberg: On the short wave asymptotic behavior of solutions of stationary problems and the asymptotic behavior as $t\rightarrow\infty$ of solutions of non-stationary problems, Russian Math Surveys, 30:2, (1975), 1-55. MR 0415085 (54:3176)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35L05

Retrieve articles in all journals with MSC (2000): 35L05


Additional Information

Jason L. Metcalfe
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: metcalfe@math.gatech.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03667-0
Received by editor(s): November 14, 2002
Published electronically: June 25, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society