## 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.## References

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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