## The defocusing energy-supercritical nonlinear wave equation in three space dimensions

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- by Rowan Killip and Monica Visan PDF
- Trans. Amer. Math. Soc.
**363**(2011), 3893-3934 Request permission

## Abstract:

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime $p>4$. For even values of the power $p$, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.## References

- Hajer Bahouri and Patrick Gérard,
*High frequency approximation of solutions to critical nonlinear wave equations*, Amer. J. Math.**121**(1999), no. 1, 131–175. MR**1705001** - J. Bourgain,
*Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case*, J. Amer. Math. Soc.**12**(1999), no. 1, 145–171. MR**1626257**, DOI 10.1090/S0894-0347-99-00283-0 - J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao,
*Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$*, Ann. of Math. (2)**167**(2008), no. 3, 767–865. MR**2415387**, DOI 10.4007/annals.2008.167.767 - J. Ginibre, A. Soffer, and G. Velo,
*The global Cauchy problem for the critical nonlinear wave equation*, J. Funct. Anal.**110**(1992), no. 1, 96–130. MR**1190421**, DOI 10.1016/0022-1236(92)90044-J - Manoussos G. Grillakis,
*Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity*, Ann. of Math. (2)**132**(1990), no. 3, 485–509. MR**1078267**, DOI 10.2307/1971427 - Manoussos G. Grillakis,
*Regularity for the wave equation with a critical nonlinearity*, Comm. Pure Appl. Math.**45**(1992), no. 6, 749–774. MR**1162370**, DOI 10.1002/cpa.3160450604 - L. V. Kapitanskiĭ,
*The Cauchy problem for the semilinear wave equation. I*, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**163**(1987), no. Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsiĭ 19, 76–104, 188 (Russian, with English summary); English transl., J. Soviet Math.**49**(1990), no. 5, 1166–1186. MR**918943**, DOI 10.1007/BF02208713 - Markus Keel and Terence Tao,
*Endpoint Strichartz estimates*, Amer. J. Math.**120**(1998), no. 5, 955–980. MR**1646048** - Carlos E. Kenig and Frank Merle,
*Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case*, Invent. Math.**166**(2006), no. 3, 645–675. MR**2257393**, DOI 10.1007/s00222-006-0011-4 - Carlos E. Kenig and Frank Merle,
*Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation*, Acta Math.**201**(2008), no. 2, 147–212. MR**2461508**, DOI 10.1007/s11511-008-0031-6 - Carlos E. Kenig and Frank Merle,
*Scattering for $\dot H^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions*, Trans. Amer. Math. Soc.**362**(2010), no. 4, 1937–1962. MR**2574882**, DOI 10.1090/S0002-9947-09-04722-9 - C. E. Kenig and F. Merle,
*Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications.*Preprint arXiv:0810.4834. - Sahbi Keraani,
*On the blow up phenomenon of the critical nonlinear Schrödinger equation*, J. Funct. Anal.**235**(2006), no. 1, 171–192. MR**2216444**, DOI 10.1016/j.jfa.2005.10.005 - R. Killip, S. Kwon, S. Shao, and M. Visan,
*On the mass-critical generalized KdV equation.*Preprint arXiv:0907.5412. - R. Killip and M. Visan,
*The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher.*To appear in Amer. J. Math. Preprint arXiv:0804.1018. - R. Killip and M. Visan,
*Nonlinear Schrödinger equations at critical regularity.*Lecture notes prepared for Clay Mathematics Institute Summer School, Zürich, Switzerland, 2008. - R. Killip and M. Visan,
*Energy-supercritical NLS: critical $H^s$-bounds imply scattering.*Preprint arXiv:0812.2084. - Rowan Killip, Terence Tao, and Monica Visan,
*The cubic nonlinear Schrödinger equation in two dimensions with radial data*, J. Eur. Math. Soc. (JEMS)**11**(2009), no. 6, 1203–1258. MR**2557134**, DOI 10.4171/JEMS/180 - Rowan Killip, Monica Visan, and Xiaoyi Zhang,
*The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher*, Anal. PDE**1**(2008), no. 2, 229–266. MR**2472890**, DOI 10.2140/apde.2008.1.229 - Cathleen S. Morawetz,
*Notes on time decay and scattering for some hyperbolic problems*, Regional Conference Series in Applied Mathematics, No. 19, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR**0492919** - 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 - Kenji Nakanishi,
*Scattering theory for the nonlinear Klein-Gordon equation with Sobolev critical power*, Internat. Math. Res. Notices**1**(1999), 31–60. MR**1666973**, DOI 10.1155/S1073792899000021 - Hartmut Pecher,
*Nonlinear small data scattering for the wave and Klein-Gordon equation*, Math. Z.**185**(1984), no. 2, 261–270. MR**731347**, DOI 10.1007/BF01181697 - Jeffrey Rauch,
*I. The $u^{5}$ Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations*, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I (Paris, 1978/1979) Res. Notes in Math., vol. 53, Pitman, Boston, Mass.-London, 1981, pp. 335–364. MR**631403** - Jalal Shatah and Michael Struwe,
*Regularity results for nonlinear wave equations*, Ann. of Math. (2)**138**(1993), no. 3, 503–518. MR**1247991**, DOI 10.2307/2946554 - Jalal Shatah and Michael Struwe,
*Geometric wave equations*, Courant Lecture Notes in Mathematics, vol. 2, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. MR**1674843** - Elias M. Stein,
*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192** - Robert S. Strichartz,
*Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations*, Duke Math. J.**44**(1977), no. 3, 705–714. MR**512086** - Michael Struwe,
*Globally regular solutions to the $u^5$ Klein-Gordon equation*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**15**(1988), no. 3, 495–513 (1989). MR**1015805** - Terence Tao,
*Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data*, New York J. Math.**11**(2005), 57–80. MR**2154347** - Terence Tao,
*Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions*, Dyn. Partial Differ. Equ.**3**(2006), no. 2, 93–110. MR**2227039**, DOI 10.4310/DPDE.2006.v3.n2.a1 - Terence Tao, Monica Visan, and Xiaoyi Zhang,
*Minimal-mass blowup solutions of the mass-critical NLS*, Forum Math.**20**(2008), no. 5, 881–919. MR**2445122**, DOI 10.1515/FORUM.2008.042 - Terence Tao, Monica Visan, and Xiaoyi Zhang,
*Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions*, Duke Math. J.**140**(2007), no. 1, 165–202. MR**2355070**, DOI 10.1215/S0012-7094-07-14015-8

## Additional Information

**Rowan Killip**- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
**Monica Visan**- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
- Received by editor(s): January 22, 2010
- Received by editor(s) in revised form: June 14, 2010
- Published electronically: January 28, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**363**(2011), 3893-3934 - MSC (2010): Primary 35L71
- DOI: https://doi.org/10.1090/S0002-9947-2011-05400-0
- MathSciNet review: 2775831