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Transactions of the American Mathematical Society

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On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles


Authors: Kunio Hidano, Jason Metcalfe, Hart F. Smith, Christopher D. Sogge and Yi Zhou
Journal: Trans. Amer. Math. Soc. 362 (2010), 2789-2809
MSC (2000): Primary 35L05, 35L20, 35L71
DOI: https://doi.org/10.1090/S0002-9947-09-05053-3
Published electronically: December 8, 2009
MathSciNet review: 2584618
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Abstract: We establish the Strauss conjecture concerning small-data global existence for nonlinear wave equations, in the setting of exterior domains to compact obstacles, for space dimensions $ n=3$ and $ 4$. The obstacle is assumed to be nontrapping, and the solution is assumed to satisfy either Dirichlet or Neumann conditions along the boundary of the obstacle. The key step in the proof is establishing certain ``abstract Strichartz estimates'' for the linear wave equation on exterior domains.


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Additional Information

Kunio Hidano
Affiliation: Department of Mathematics, Mie University, Mie Prefecture, Japan

Jason Metcalfe
Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3260

Hart F. Smith
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350

Christopher D. Sogge
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21244

Yi Zhou
Affiliation: School of Mathematical Science, Fudan University, Shanghai, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9947-09-05053-3
Received by editor(s): February 13, 2009
Published electronically: December 8, 2009
Additional Notes: The first author was supported in part by the Grant-in-Aid for Young Scientists (B) (No. 18740069), The Ministry of Education, Culture, Sports, Science and Technology, Japan, and he would like to thank the Department of Mathematics at the Johns Hopkins University for the hospitality and financial support during his visit where part of this research was carried out
The second, third and fourth authors were supported by the National Science Foundation
The fifth author was supported by project 10728101 of NSFC and the “111” project and Doctoral Programme Foundation of the Ministry of Education of China.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.