The radial defocusing energysupercritical nonlinear wave equation in all space dimensions
Authors:
Rowan Killip and Monica Visan
Journal:
Proc. Amer. Math. Soc. 139 (2011), 18051817
MSC (2010):
Primary 35L71
Published electronically:
November 1, 2010
MathSciNet review:
2763767
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Abstract: We consider the defocusing nonlinear wave equation with sphericallysymmetric initial data in the regime (which is energysupercritical) and dimensions ; we also consider , but for a smaller range of . The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximallifespan solutions with bounded critical Sobolev norm are global and scatter.
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 H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121 (1999), 131175. MR 1705001
 2.
 J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Amer. Math. Soc. 12 (1999), 145171. MR 1626257
 3.
 A. Bulut, Maximizers for the Strichartz inequalities for the wave equation. Preprint arXiv:0905.1678.
 4.
 A. Bulut, Global wellposedness and scattering for the defocusing energysupercritical cubic nonlinear wave equation. Preprint arXiv:1006.4168
 5.
 A. Bulut, M. Czubak, D. Li, N. Pavlovic, X. Zhang, Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions. Preprint arXiv:0911.4534.
 6.
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 7.
 M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. of Math. (2) 132 (1990), 485509. MR 1078267
 8.
 M. Grillakis, Regularity for the wave equation with a critical nonlinearity. Comm. Pure Appl. Math. 45 (1992), 749774. MR 1162370 (93e:35073)
 9.
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 10.
 M. Keel and T. Tao, Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), 955980. MR 1646048
 11.
 C. E. Kenig and F. Merle, Global wellposedness, scattering and blowup for the energycritical, focusing, nonlinear Schrödinger equation in the radial case. Invent. Math. 166 (2006), 645675. MR 2257393
 12.
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 13.
 C. E. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical nonlinear wave equations, with applications. Preprint arXiv:0810.4834.
 14.
 S. Keraani, On the blowup phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235 (2006), 171192. MR 2216444
 15.
 R. Killip and M. Visan, The focusing energycritical nonlinear Schrödinger equation in dimensions five and higher. Amer. J. Math. 132 (2010), 361424. MR 2654778
 16.
 R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity. Lecture notes prepared for Clay Mathematics Institute Summer School, Zürich, Switzerland, 2008.
 17.
 R. Killip and M. Visan, Energysupercritical NLS: critical bounds imply scattering. Preprint arXiv:0812.2084. To appear in Comm. Partial Differential Equations.
 18.
 R. Killip, M. Visan, and X. Zhang, Energycritical NLS with quadratic potentials. Comm. Partial Differential Equations 34 (2009), 15311565. MR 2581982
 19.
 R. Killip and M. Visan, The defocusing energysupercritical nonlinear wave equation in three space dimensions. Preprint http://arxiv.org/abs/1001.1761arXiv:1001.1761. To appear in Trans. Amer. Math. Soc.
 20.
 C. 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
 21.
 C. S. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation. Comm. Pure Appl. Math. 25 (1972), 131. MR 0303097
 22.
 K. Nakanishi, Scattering theory for nonlinear KleinGordon equation with Sobolev critical power. Internat. Math. Res. Notices 1 (1999), 3160. MR 1666973
 23.
 H. Pecher, Nonlinear small data scattering for the wave and KleinGordon equation. Math. Z. 185 (1984), 261270. MR 0731347
 24.
 J. Rauch, I. The KleinGordon equation. II. Anomalous singularities for semilinear wave equations. In ``Nonlinear partial differential equations and their applications''. Collège de France Seminar, Vol. I (Paris, 1978/1979), pp. 335364, Res. Notes in Math., 53, Pitman, Boston, Mass.London, 1981. MR 0631403
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 J. Shatah and M. Struwe, Regularity results for nonlinear wave equations. Ann. of Math. (2) 138 (1993), 503518. MR 1247991
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 J. Shatah and M. Struwe, Geometric wave equations. Courant Lecture Notes in Mathematics, 2. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. MR 1674843
 27.
 R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), 705714. MR 0512086
 28.
 M. Struwe, Globally regular solutions to the KleinGordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1988), 495513. MR 1015805
 29.
 T. Tao, Global wellposedness and scattering for the higherdimensional energycritical nonlinear Schrödinger equation for radial data. New York J. of Math. 11 (2005), 5780. MR 2154347
 30.
 T. Tao, Spacetime bounds for the energycritical nonlinear wave equation in three spatial dimensions. Dynamics of Partial Differential Equations 3 (2006), 93110. MR 2227039
 31.
 M. E. Taylor, Tools for PDE. Mathematical Surveys and Monographs, 81. American Mathematical Society, Providence, RI, 2000. MR 1766415
 32.
 M. Visan, The defocusing energycritical nonlinear Schrödinger equation in higher dimensions. Duke Math. J. 138 (2007), 281374. MR 2318286
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Additional Information
Rowan Killip
Affiliation:
Department of Mathematics, University of California, Los Angeles, California 900951555
Monica Visan
Affiliation:
Department of Mathematics, University of California, Los Angeles, California 900951555
DOI:
http://dx.doi.org/10.1090/S000299392010106159
Received by editor(s):
February 8, 2010
Received by editor(s) in revised form:
May 26, 2010
Published electronically:
November 1, 2010
Additional Notes:
The first author was supported by NSF grant DMS0701085
The second author was supported by NSF grant DMS0901166
Communicated by:
Hart F. Smith
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
