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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Almost global existence for quasilinear wave equations in three space dimensions
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by Markus Keel, Hart F. Smith and Christopher D. Sogge PDF
J. Amer. Math. Soc. 17 (2004), 109-153 Request permission

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 $O(|x|^{-1})$ decay of solutions of the wave equation as much as the $O(|t|^{-1})$ 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 $O(1/|x|)$-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
  • MR Author ID: 164510
  • Received by editor(s): September 16, 2002
  • Published electronically: September 30, 2003
  • Additional Notes: The authors were supported in part by the NSF
  • © Copyright 2003 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 109-153
  • MSC (2000): Primary 35L05, 35L10, 35L15, 35L20, 35L70
  • DOI: https://doi.org/10.1090/S0894-0347-03-00443-0
  • MathSciNet review: 2015331