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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
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Abstract:

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space $\mathbb H^n$ 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 $\mathbb R^{n} \times \mathbb R$ which extend the ones of Georgiev, Lindblad, and Sogge.


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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

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