Radial solutions to energy supercritical wave equations in odd dimensions

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December 2011, 31(4): 1365-1381. doi: 10.3934/dcds.2011.31.1365

Radial solutions to energy supercritical wave equations in odd dimensions

Carlos E. Kenig ^1, and Frank Merle ^2,

1.	Department of Mathematics, University of Chicago, Chicago, Illinois, 60637-1514, United States
2.	Cergy Pontoise (UMR 8088) and IHES, France

Received March 2009 Revised April 2010 Published September 2011

Abstract
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We establish pointwise decay bounds for radial, compact solutions of energy supercritical wave equations in odd dimensions. Applications are given.

Keywords: decay estimates., odd dimensions, Energy supercritical, radial compact solutions.

Mathematics Subject Classification: Primary: 35L70; Secondary: 35Q5.

Citation: Carlos E. Kenig, Frank Merle. Radial solutions to energy supercritical wave equations in odd dimensions. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1365-1381. doi: 10.3934/dcds.2011.31.1365

References:

[1]	H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. doi: 10.1353/ajm.1999.0001.
[2]	A. Bulut, Maximizers for the Strichartz inequality for the wave equation, preprint, arXiv:0905.1678.
[3]	A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation, preprint, arXiv:1006.4168.
[4]	A. Bulut, M. Czubak, D. Li, N. Pavlovic and X. Zhang, Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions, preprint, arXiv:0911.4534.
[5]	L. Iskauriaza, G. Serëgin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44; translation in Russian Math. Surveys, 58 (2003), 211-250.
[6]	I. Gallagher, G. Koch and F. Planchon, A profile decomposition approach to the $L^\infty_tL^3_x$ Navier-Stokes regularity criterion, preprint, arXiv:1012.0145.
[7]	J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68. doi: 10.1006/jfan.1995.1119.
[8]	C. Kenig, Global well-posedness and scattering for the energy critical focusing nonlinear Schrödinger and wave equations, Lecture notes for a mini course given at "Analyse des Equations aux Derivées Partielles," Evian-les-Bains, 2007. Available from: http://www.math.uchicago.edu/~cek.
[9]	C. Kenig, The concentration-compactness/rigidity theorem method for critical dispersive and wave equations, Lectures for a course given at CRM, Bellaterra, Spain, May 2008. Available from: http://www.math.uchicago.edu/~cek.
[10]	C. Kenig, Recent developments on the global behavior to critical nonlinear dispersive equations, Proceedings of the International Congress of Mathematicians 2010, 1 (2010), 326-338.
[11]	C. Kenig and G. Koch, An alternative approach to regularity for the Navier-Stokes equation in a critical space, to appear on Ann. Inst. H. Poincaré Anal. Non Linéaire, 2011.
[12]	C. Kenig and F. 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), 645-675. doi: 10.1007/s00222-006-0011-4.
[13]	C. Kenig and F. Merle, Global well-posedness, scatering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.
[14]	C. Kenig and F. Merle, Scattering for $\dot H^{1/2}$ bounded solutions to the cubic defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962. doi: 10.1090/S0002-9947-09-04722-9.
[15]	C. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, to appear on Amer. J. Math., 2011.
[16]	C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[17]	R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions, preprint, arXiv:1001.1761.
[18]	R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions, preprint, arXiv:1002.1756.
[19]	F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, 1998, 399-425. doi: 10.1155/S1073792898000270.
[20]	C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A, 306 (1968), 291-296. doi: 10.1098/rspa.1968.0151.
[21]	B. Perthame and L. Vega, Morrey-Campanato estimates for Helmholtz equations, J. Funct. Anal., 164 (1999), 340-355. doi: 10.1006/jfan.1999.3391.
[22]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970.
[23]	R. J. Taggart, Inhomogeneous Stichartz estimates, Forum Math., 22 (2010), 825-853. doi: 10.1515/FORUM.2010.044.

show all references

References:

[1]	H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. doi: 10.1353/ajm.1999.0001.
[2]	A. Bulut, Maximizers for the Strichartz inequality for the wave equation, preprint, arXiv:0905.1678.
[3]	A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation, preprint, arXiv:1006.4168.
[4]	A. Bulut, M. Czubak, D. Li, N. Pavlovic and X. Zhang, Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions, preprint, arXiv:0911.4534.
[5]	L. Iskauriaza, G. Serëgin and V. Shverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44; translation in Russian Math. Surveys, 58 (2003), 211-250.
[6]	I. Gallagher, G. Koch and F. Planchon, A profile decomposition approach to the $L^\infty_tL^3_x$ Navier-Stokes regularity criterion, preprint, arXiv:1012.0145.
[7]	J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68. doi: 10.1006/jfan.1995.1119.
[8]	C. Kenig, Global well-posedness and scattering for the energy critical focusing nonlinear Schrödinger and wave equations, Lecture notes for a mini course given at "Analyse des Equations aux Derivées Partielles," Evian-les-Bains, 2007. Available from: http://www.math.uchicago.edu/~cek.
[9]	C. Kenig, The concentration-compactness/rigidity theorem method for critical dispersive and wave equations, Lectures for a course given at CRM, Bellaterra, Spain, May 2008. Available from: http://www.math.uchicago.edu/~cek.
[10]	C. Kenig, Recent developments on the global behavior to critical nonlinear dispersive equations, Proceedings of the International Congress of Mathematicians 2010, 1 (2010), 326-338.
[11]	C. Kenig and G. Koch, An alternative approach to regularity for the Navier-Stokes equation in a critical space, to appear on Ann. Inst. H. Poincaré Anal. Non Linéaire, 2011.
[12]	C. Kenig and F. 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), 645-675. doi: 10.1007/s00222-006-0011-4.
[13]	C. Kenig and F. Merle, Global well-posedness, scatering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.
[14]	C. Kenig and F. Merle, Scattering for $\dot H^{1/2}$ bounded solutions to the cubic defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962. doi: 10.1090/S0002-9947-09-04722-9.
[15]	C. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, to appear on Amer. J. Math., 2011.
[16]	C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[17]	R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions, preprint, arXiv:1001.1761.
[18]	R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions, preprint, arXiv:1002.1756.
[19]	F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, 1998, 399-425. doi: 10.1155/S1073792898000270.
[20]	C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A, 306 (1968), 291-296. doi: 10.1098/rspa.1968.0151.
[21]	B. Perthame and L. Vega, Morrey-Campanato estimates for Helmholtz equations, J. Funct. Anal., 164 (1999), 340-355. doi: 10.1006/jfan.1999.3391.
[22]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970.
[23]	R. J. Taggart, Inhomogeneous Stichartz estimates, Forum Math., 22 (2010), 825-853. doi: 10.1515/FORUM.2010.044.

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