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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Almost global existence for quasilinear wave equations in three space dimensions


Authors: Markus Keel, Hart F. Smith and Christopher D. Sogge
Journal: J. Amer. Math. Soc. 17 (2004), 109-153
MSC (2000): Primary 35L05, 35L10, 35L15, 35L20, 35L70
Published electronically: September 30, 2003
MathSciNet review: 2015331
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Abstract: We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only use the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(\vert x\vert^{-1})$ decay of solutions of the wave equation as much as the $O(\vert t\vert^{-1})$ decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equation that gives $O(1/t)$ decay of solutions of the inhomogeneous linear wave equation in terms of a $O(1/\vert x\vert)$-weighted norm on the forcing term. A weighted $L^{2}$ space-time estimate for inhomogeneous wave equations is also important in making the spatial decay useful for the long-term existence argument.


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Additional Information

Markus Keel
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

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

Christopher D. Sogge
Affiliation: Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218

DOI: http://dx.doi.org/10.1090/S0894-0347-03-00443-0
PII: S 0894-0347(03)00443-0
Received by editor(s): September 16, 2002
Published electronically: September 30, 2003
Additional Notes: The authors were supported in part by the NSF
Article copyright: © Copyright 2003 American Mathematical Society