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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

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 & 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.  Google Scholar

[2]

A. Bulut, Maximizers for the Strichartz inequality for the wave equation,, preprint, ().   Google Scholar

[3]

A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation,, preprint, ().   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[6]

I. Gallagher, G. Koch and F. Planchon, A profile decomposition approach to the $L^\infty_tL^3_x$ Navier-Stokes regularity criterion,, preprint, ().   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions,, preprint, ().   Google Scholar

[18]

R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions,, preprint, ().   Google Scholar

[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.  doi: 10.1155/S1073792898000270.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970.  Google Scholar

[23]

R. J. Taggart, Inhomogeneous Stichartz estimates, Forum Math., 22 (2010), 825-853. doi: 10.1515/FORUM.2010.044.  Google Scholar

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.  Google Scholar

[2]

A. Bulut, Maximizers for the Strichartz inequality for the wave equation,, preprint, ().   Google Scholar

[3]

A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation,, preprint, ().   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[6]

I. Gallagher, G. Koch and F. Planchon, A profile decomposition approach to the $L^\infty_tL^3_x$ Navier-Stokes regularity criterion,, preprint, ().   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions,, preprint, ().   Google Scholar

[18]

R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions,, preprint, ().   Google Scholar

[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.  doi: 10.1155/S1073792898000270.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970.  Google Scholar

[23]

R. J. Taggart, Inhomogeneous Stichartz estimates, Forum Math., 22 (2010), 825-853. doi: 10.1515/FORUM.2010.044.  Google Scholar

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