## 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
- J. Amer. Math. Soc.
**17**(2004), 109-153 - DOI: https://doi.org/10.1090/S0894-0347-03-00443-0
- Published electronically: September 30, 2003
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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 $O(|x|^{-1})$ 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

- Demetrios Christodoulou,
*Global solutions of nonlinear hyperbolic equations for small initial data*, Comm. Pure Appl. Math.**39**(1986), no. 2, 267–282. MR**820070**, DOI 10.1002/cpa.3160390205 - P. S. Datti,
*Nonlinear wave equations in exterior domains*, Nonlinear Anal.**15**(1990), no. 4, 321–331. MR**1066388**, DOI 10.1016/0362-546X(90)90140-C - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - Paul Godin,
*Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems*, Amer. J. Math.**117**(1995), no. 6, 1475–1505. MR**1363076**, DOI 10.2307/2375027 - Nakao Hayashi,
*Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain*, J. Funct. Anal.**131**(1995), no. 2, 302–344. MR**1345034**, DOI 10.1006/jfan.1995.1091 - Lars Hörmander,
*Lectures on nonlinear hyperbolic differential equations*, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR**1466700** - Lars Hörmander,
*$L^1,\ L^\infty$ estimates for the wave operator*, Analyse mathématique et applications, Gauthier-Villars, Montrouge, 1988, pp. 211–234. MR**956961** - Fritz John,
*Nonlinear wave equations, formation of singularities*, University Lecture Series, vol. 2, American Mathematical Society, Providence, RI, 1990. Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. MR**1066694**, DOI 10.1090/ulect/002 - F. John and S. Klainerman,
*Almost global existence to nonlinear wave equations in three space dimensions*, Comm. Pure Appl. Math.**37**(1984), no. 4, 443–455. MR**745325**, DOI 10.1002/cpa.3160370403 - Markus Keel, Hart F. Smith, and Christopher D. Sogge,
*Global existence for a quasilinear wave equation outside of star-shaped domains*, J. Funct. Anal.**189**(2002), no. 1, 155–226. MR**1887632**, DOI 10.1006/jfan.2001.3844
KSS2 M. Keel, H. Smith, and C. D. Sogge: - Sergiu Klainerman,
*Uniform decay estimates and the Lorentz invariance of the classical wave equation*, Comm. Pure Appl. Math.**38**(1985), no. 3, 321–332. MR**784477**, DOI 10.1002/cpa.3160380305 - S. Klainerman,
*The null condition and global existence to nonlinear wave equations*, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR**837683** - Sergiu Klainerman and Thomas C. Sideris,
*On almost global existence for nonrelativistic wave equations in $3$D*, Comm. Pure Appl. Math.**49**(1996), no. 3, 307–321. MR**1374174**, DOI 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H - Mikhail Kovalyov,
*Long-time behaviour of solutions of a system of nonlinear wave equations*, Comm. Partial Differential Equations**12**(1987), no. 5, 471–501. MR**883321**, DOI 10.1080/03605308708820501 - P. D. Lax, C. S. Morawetz, and R. S. Phillips,
*Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle*, Comm. Pure Appl. Math.**16**(1963), 477–486. MR**155091**, DOI 10.1002/cpa.3160160407 - Peter D. Lax and Ralph S. Phillips,
*Scattering theory*, 2nd ed., Pure and Applied Mathematics, vol. 26, Academic Press, Inc., Boston, MA, 1989. With appendices by Cathleen S. Morawetz and Georg Schmidt. MR**1037774** - Cathleen S. Morawetz,
*The decay of solutions of the exterior initial-boundary value problem for the wave equation*, Comm. Pure Appl. Math.**14**(1961), 561–568. MR**132908**, DOI 10.1002/cpa.3160140327 - Cathleen S. Morawetz,
*Time decay for the nonlinear Klein-Gordon equations*, Proc. Roy. Soc. London Ser. A**306**(1968), 291–296. MR**234136**, DOI 10.1098/rspa.1968.0151 - Cathleen S. Morawetz and Walter A. Strauss,
*Decay and scattering of solutions of a nonlinear relativistic wave equation*, Comm. Pure Appl. Math.**25**(1972), 1–31. MR**303097**, DOI 10.1002/cpa.3160250103 - Cathleen S. Morawetz, James V. Ralston, and Walter A. Strauss,
*Decay of solutions of the wave equation outside nontrapping obstacles*, Comm. Pure Appl. Math.**30**(1977), no. 4, 447–508. MR**509770**, DOI 10.1002/cpa.3160300405 - Thomas C. Sideris,
*Nonresonance and global existence of prestressed nonlinear elastic waves*, Ann. of Math. (2)**151**(2000), no. 2, 849–874. MR**1765712**, DOI 10.2307/121050 - Thomas C. Sideris,
*The null condition and global existence of nonlinear elastic waves*, Invent. Math.**123**(1996), no. 2, 323–342. MR**1374204**, DOI 10.1007/s002220050030 - Thomas C. Sideris and Shu-Yi Tu,
*Global existence for systems of nonlinear wave equations in 3D with multiple speeds*, SIAM J. Math. Anal.**33**(2001), no. 2, 477–488. MR**1857981**, DOI 10.1137/S0036141000378966 - Hart F. Smith and Christopher D. Sogge,
*Global Strichartz estimates for nontrapping perturbations of the Laplacian*, Comm. Partial Differential Equations**25**(2000), no. 11-12, 2171–2183. MR**1789924**, DOI 10.1080/03605300008821581 - Christopher D. Sogge,
*Lectures on nonlinear wave equations*, Monographs in Analysis, II, International Press, Boston, MA, 1995. MR**1715192**
So2 C. D. Sogge: - Yoshihiro Shibata and Yoshio Tsutsumi,
*On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain*, Math. Z.**191**(1986), no. 2, 165–199. MR**818663**, DOI 10.1007/BF01164023 - Kazuyoshi Yokoyama,
*Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions*, J. Math. Soc. Japan**52**(2000), no. 3, 609–632. MR**1760608**, DOI 10.2969/jmsj/05230609

*Almost global existence for some semilinear wave equations*, J. d’Anal. Math.

**87**(2002), 265–279.

*Global existence for nonlinear wave equations with multiple speeds*, Proceedings of the 2001 Mount Holyoke Conference on Harmonic Analysis, pp. 353–366, Contemp. Math.

**320**, Amer. Math. Soc., Providence, RI, 2003.

## Bibliographic 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