The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

Killip, Rowan; Visan, Monica

doi:10.1090/S0002-9939-2010-10615-9

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions
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by Rowan Killip and Monica Visan PDF

Proc. Amer. Math. Soc. 139 (2011), 1805-1817 Request permission

Abstract:

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac 4{d-2}<p<\frac 4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also consider $d\geq 7$, but for a smaller range of $p>\frac 4{d-2}$. 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 maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.

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
A. Bulut, Maximizers for the Strichartz inequalities for the wave equation. Preprint arXiv:0905.1678.
A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation. Preprint arXiv:1006.4168
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.
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
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
Rowan Killip and Monica Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math. 132 (2010), no. 2, 361–424. MR 2654778, DOI 10.1353/ajm.0.0107
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. To appear in Comm. Partial Differential Equations.
Rowan Killip, Monica Visan, and Xiaoyi Zhang, Energy-critical NLS with quadratic potentials, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1531–1565. MR 2581982, DOI 10.1080/03605300903328109
R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions. Preprint arXiv:1001.1761. To appear in Trans. Amer. Math. Soc.
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
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
Michael E. Taylor, Tools for PDE, Mathematical Surveys and Monographs, vol. 81, American Mathematical Society, Providence, RI, 2000. Pseudodifferential operators, paradifferential operators, and layer potentials. MR 1766415, DOI 10.1090/surv/081
Monica Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 (2007), no. 2, 281–374. MR 2318286, DOI 10.1215/S0012-7094-07-13825-0