Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

Author:
Daniel Tataru

Journal:
Trans. Amer. Math. Soc. **353** (2001), 795-807

MSC (2000):
Primary 35L05, 35L70; Secondary 58J45

DOI:
https://doi.org/10.1090/S0002-9947-00-02750-1

Published electronically:
October 23, 2000

MathSciNet review:
1804518

Full-text PDF

Abstract | References | Similar Articles | Additional Information

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in which extend the ones of Georgiev, Lindblad, and Sogge.

**1.**J. Bergh and J. Löfström,*Interpolation Spaces: An Introduction*, Springer-Verlag, 1976 MR**58:2349****2.**W.O. Bray,*Aspects of Harmonic Analysis on Real Hyperbolic Space Functional Analysis*, (Orono, ME, 1992), Lecture Notes in Pure and Applied Mathematics 157, Marcel Dekker, 1994, pp. 77-102. MR**95k:43008****3.**P. Brenner,*On - estimates for the wave equations*, Math. Z. 145(1975), 251-254 MR**52:8658****4.**J. Clerc and E.M. Stein,*multipliers for noncompact symmetric spaces*, Proc. Nat. Acad. Sci. USA 71(1974), 3911-3912 MR**51:3803****5.**R. Glassey,*Existence in the large for in two space dimensions*, Math. Z. 178(1981), 233-261 MR**84h:35106****6.**J. Ginibre and G. Velo,*Generalized Strichartz inequality for the wave equation*, J. Funct. Anal. 133 (1995), 50-68. MR**97a:46047****7.**V. Georgiev, H. Lindblad and C. Sogge,*Weighted Strichartz estimates and global existence for semilinear wave equations*, Amer. J. Math. 119 (1997), 1291-1319 MR**99f:35134****8.**M. Grillakis,*Regularity and asymptotic behavior of the wave equation with a critical nonlinearity*, Ann. of Math. 132(1990), 485-509 MR**92c:35080****9.**M. Grillakis,*Regularity for the wave equation with a critical nonlinearity*, Comm. Pure Appl. Math 45 (1992) 749-774 MR**93e:35073****10.**S. Helgason,*Groups and Geometric Analysis*, Academic Press 1984 MR**86c:22017****11.**F. John,*Blow-up of solutions of nonlinear wave equations in three space dimensions*, Manuscripta Math 28 (1979), 235-265 MR**80i:35114****12.**H. Kubo,*On the critical decay and power for semilinear wave equations in odd space dimensions*, Discrete Contin. Dynam. Systems 2 (1996), 173-190 MR**96m:35221****13.**H. Lindblad and C. Sogge,*On existence and scattering with minimal regularity for semilinear wave equations*,*J Funct. Anal.*, 1995, 357-426 MR**96i:35087****14.**H. Lindblad and C. Sogge,*Long-time existence for small amplitude semilinear wave equation*, Amer. J. Math. 118 (1996), 1047-1135 MR**97h:35158****15.**H. Pecher*Non-linear small data scattering for the wave and Klein-Gordon equations*, Math. Z. 185(1984), 261-270 MR**85h:35165****16.**I. Segal,*Space-time decay for solutions of wave equations*, Adv. in Math. 22(1976), 305-311 MR**58:11945****17.**T. Sideris,*Nonexistence of global solutions to semilinear wave equations in high dimensions*, Comm. PDE 12 (1987), 378-406 MR**86d:35090****18.**E.M. Stein,*Interpolation of linear operators*, Trans. Amer. Math. Soc. 83 (1956), 482-492 MR**18:575d****19.**E.M. Stein,*Harmonic Analysis*, Princeton University Press, 1993 MR**95c:42002****20.**R.S. Strichartz,*Restrictions of Fourier transform to quadratic surfaces and decay of solutions of Wave Equations*, Duke Math. J. 44(1977) 705-714 MR**58:23577****21.**Y. Zhou,*Cauchy problem for semilinear wave equations in four space dimensions with small initial data*, J. Partial Diff. Equations 8 (1995) 135-144 MR**96c:35128**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35L05,
35L70,
58J45

Retrieve articles in all journals with MSC (2000): 35L05, 35L70, 58J45

Additional Information

**Daniel Tataru**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08540

Address at time of publication:
Department of Mathematics, Northwestern University, Evanston, Illinois 08540

Email:
tataru@math.nwu.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02750-1

Keywords:
Semilinear wave equation,
Strichartz estimates,
hyperbolic space

Received by editor(s):
October 10, 1997

Published electronically:
October 23, 2000

Additional Notes:
Research partially supported by NSF grant DMS-9622942 and by an Alfred P. Sloan fellowship

Article copyright:
© Copyright 2000
American Mathematical Society