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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
DOI: https://doi.org/10.1090/S0894-0347-03-00443-0
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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  • 1. D. Christodoulou: Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39, (1986), 267-282. MR 87c:35111
  • 2. P. S. Datti: Nonlinear wave equations in exterior domains, Nonlinear Anal. 15 (1990), 321-331. MR 91k:35163
  • 3. D. Gilbarg and N. Trudinger: Elliptic partial differential equations of second order, Springer-Verlag, Second edition, Third Printing, 1998. MR 86c:35035
  • 4. P. Godin: Global existence of solutions to some exterior radial quasilinear Cauchy-Dirichlet problems, Amer. J. Math. 117 (1995), 1475-1505. MR 97j:35096
  • 5. N. Hayashi: Global existence of small solutions to quadratic nonlinear wave equatinos in an exterior domain, J. Funct. Anal. 131 (1995), 302-344. MR 96f:35113
  • 6. L. Hörmander: Lectures on nonlinear hyperbolic equations, Springer-Verlag, Berlin, 1997. MR 98e:35103
  • 7. L. Hörmander: $L^1, L^\infty$ estimates for the wave operator, Analyse Mathématique et Applications, Gauthier-Villars, Montrouge, 1988, pp. 211-234. MR 90e:35113
  • 8. F. John: Nonlinear wave equations, formation of singularities, American Mathematical Society, Providence, RI, 1989. MR 91g:35001
  • 9. F. John and S. Klainerman: Almost global existence to nonlinear wave equations in three dimensions Comm. Pure Appl. Math. 37 (1984), 443-455. MR 85k:35147
  • 10. M. Keel, H. Smith, and C. D. Sogge: Global existence for a quasilinear wave equation outside of star-shaped domains, J. Funct. Anal. 189 (2002), 155-226. MR 2003f:35209
  • 11. M. Keel, H. Smith, and C. D. Sogge: Almost global existence for some semilinear wave equations, J. d'Anal. Math. 87 (2002), 265-279.
  • 12. S. Klainerman: Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), 321-332. MR 86i:35091
  • 13. S. Klainerman: The null condition and global existence to nonlinear wave equations, Lectures in Applied Math. 23 (1986), 293-326. MR 87h:35217
  • 14. S. Klainerman and T. Sideris: On almost global existence for nonrelativistic wave equations in $3$D, Comm. Pure Appl. Math. 49 (1996), 307-321. MR 96m:35231
  • 15. M. Kovalyov: Long-time behavior of solutions of a system of nonlinear wave equations, Comm. Partial Differential Equations 12 (1987), 471-501. MR 88c:35100
  • 16. 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 27:5033
  • 17. P. D. Lax and R. S. Phillips: Scattering theory, revised edition, Academic Press, San Diego, 1989. MR 90k:35005
  • 18. C. S. Morawetz: The decay of solutions of the exterior initial-boundary problem for the wave equation, Comm. Pure Appl. Math. 14 (1961), 561-568. MR 24:A2744
  • 19. C. S. Morawetz: Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. A 306 (1968), 291-296. MR 38:2455
  • 20. C. S. Morawetz and W. Strauss: Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1-31. MR 46:2239
  • 21. C. S. Morawetz, J. Ralston, and W. Strauss: Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math. 30 (1977), 447-508. MR 58:23091a
  • 22. T. Sideris: Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. 151 (2000), 849-874. MR 2001g:35180
  • 23. T. Sideris: The null condition and global existence of nonlinear elastic waves, Invent. Math. 123 (1996), 323-342. MR 97a:35158
  • 24. T. Sideris and S. Y. Tu: Global existence for systems of nonlinear wave equations in $3$D with multiple speeds, SIAM J. Math. Anal. 33 (2001), 477-488. MR 2002j:35220
  • 25. H. Smith and C. D. Sogge: Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), 2171-2183. MR 2001j:35180
  • 26. C. D. Sogge: Lectures on nonlinear wave equations, International Press, Boston, MA, 1995. MR 2000g:35153
  • 27. C. D. Sogge: 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.
  • 28. Y. Shibata and Y. Tsutsumi: On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z. 191 (1986), 165-199. MR 87i:35122
  • 29. K. Yokoyama: Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. 52 (2000), 609-632. MR 2002a:35156

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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: https://doi.org/10.1090/S0894-0347-03-00443-0
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

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