June  2013, 33(6): 2423-2450. doi: 10.3934/dcds.2013.33.2423

Global dynamics of the nonradial energy-critical wave equation above the ground state energy

1. 

Bâtiment des Mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland

2. 

Department of Mathematics, Kyoto University, Kyoto 606-8502

3. 

Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60615, United States

Received  January 2012 Revised  August 2012 Published  December 2012

In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions $3$ and $5$ assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in $\dot H^{1}\times L^{2}$ with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as $t → ±∞$.
Citation: Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag. Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2423-2450. doi: 10.3934/dcds.2013.33.2423
References:
[1]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, Amer. J. Math., 121 (1999), 131.   Google Scholar

[2]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation,, J. Eur. Math. Soc., 13 (2011), 533.  doi: 10.4171/JEMS/261.  Google Scholar

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T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case,, preprint, ().  doi: 10.4171/JEMS/336.  Google Scholar

[5]

T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, preprint, ().  doi: 10.1007/s00039-012-0174-7.  Google Scholar

[6]

T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, preprint, ().   Google Scholar

[7]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS,, Geom. Funct. Anal., 18 (2009), 1787.  doi: 10.1007/s00039-009-0707-x.  Google Scholar

[8]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical wave equation,, Int. Math. Res. Pap. IMRP, (2008).   Google Scholar

[9]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical non-linear wave equation,, J. Funct. Anal., 110 (1992), 96.  doi: 10.1016/0022-1236(92)90044-J.  Google Scholar

[10]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation,, Anal. PDE, 4 (2011), 405.  doi: 10.2140/apde.2011.4.405.  Google Scholar

[11]

C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case,, Invent. Math., 166 (2006), 645.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[12]

C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation,, Acta Math., 201 (2008), 147.  doi: 10.1007/s11511-008-0031-6.  Google Scholar

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation,, to appear in Amer. Journal Math., ().   Google Scholar

[14]

J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation,, Amer. J. Math., 129 (2007), 843.  doi: 10.1353/ajm.2007.0021.  Google Scholar

[15]

J. Krieger, W. Schlag and D. Tataru, Slow blow-up solutions for the $H^1(\mathbbR^3)$ critical focusing semilinear wave equation,, Duke Math. J., 147 (2009), 1.  doi: 10.1215/00127094-2009-005.  Google Scholar

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I,, Rev. Mat. Iberoamericana, 1 (1985), 145.  doi: 10.4171/RMI/6.  Google Scholar

[17]

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

[18]

K. Nakanishi, Scattering theory for the nonlinear Klein-Gordon equation with Sobolev critical power,, Internat. Math. Res. Notices, 1999 (): 31.  doi: 10.1155/S1073792899000021.  Google Scholar

[19]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation,, Journal Diff. Eq., 250 (2011), 2299.  doi: 10.1016/j.jde.2010.10.027.  Google Scholar

[20]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D,, Calc. Var. and PDE, 44 (2012), 1.  doi: 10.1007/s00526-011-0424-9.  Google Scholar

[21]

K. Nakanishi and W. Schlag, Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption,, Arch. Rational Mech. Analysis, 203 (2012), 809.  doi: 10.1007/s00205-011-0462-7.  Google Scholar

[22]

K. Nakanishi and W. Schlag, "Invariant Manifolds and Dispersive Hamiltonian Evolution Equations,", Zürich Lectures in Advanced Mathematics, (2011).  doi: 10.4171/095.  Google Scholar

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.   Google Scholar

[24]

J. Shatah and M. Struwe, "Geometric Wave Equations,", Courant Lecture Notes, (1998).   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.   Google Scholar

[2]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation,, J. Eur. Math. Soc., 13 (2011), 533.  doi: 10.4171/JEMS/261.  Google Scholar

[4]

T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case,, preprint, ().  doi: 10.4171/JEMS/336.  Google Scholar

[5]

T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, preprint, ().  doi: 10.1007/s00039-012-0174-7.  Google Scholar

[6]

T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, preprint, ().   Google Scholar

[7]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS,, Geom. Funct. Anal., 18 (2009), 1787.  doi: 10.1007/s00039-009-0707-x.  Google Scholar

[8]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical wave equation,, Int. Math. Res. Pap. IMRP, (2008).   Google Scholar

[9]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical non-linear wave equation,, J. Funct. Anal., 110 (1992), 96.  doi: 10.1016/0022-1236(92)90044-J.  Google Scholar

[10]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation,, Anal. PDE, 4 (2011), 405.  doi: 10.2140/apde.2011.4.405.  Google Scholar

[11]

C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case,, Invent. Math., 166 (2006), 645.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[12]

C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation,, Acta Math., 201 (2008), 147.  doi: 10.1007/s11511-008-0031-6.  Google Scholar

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation,, to appear in Amer. Journal Math., ().   Google Scholar

[14]

J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation,, Amer. J. Math., 129 (2007), 843.  doi: 10.1353/ajm.2007.0021.  Google Scholar

[15]

J. Krieger, W. Schlag and D. Tataru, Slow blow-up solutions for the $H^1(\mathbbR^3)$ critical focusing semilinear wave equation,, Duke Math. J., 147 (2009), 1.  doi: 10.1215/00127094-2009-005.  Google Scholar

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I,, Rev. Mat. Iberoamericana, 1 (1985), 145.  doi: 10.4171/RMI/6.  Google Scholar

[17]

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

[18]

K. Nakanishi, Scattering theory for the nonlinear Klein-Gordon equation with Sobolev critical power,, Internat. Math. Res. Notices, 1999 (): 31.  doi: 10.1155/S1073792899000021.  Google Scholar

[19]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation,, Journal Diff. Eq., 250 (2011), 2299.  doi: 10.1016/j.jde.2010.10.027.  Google Scholar

[20]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D,, Calc. Var. and PDE, 44 (2012), 1.  doi: 10.1007/s00526-011-0424-9.  Google Scholar

[21]

K. Nakanishi and W. Schlag, Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption,, Arch. Rational Mech. Analysis, 203 (2012), 809.  doi: 10.1007/s00205-011-0462-7.  Google Scholar

[22]

K. Nakanishi and W. Schlag, "Invariant Manifolds and Dispersive Hamiltonian Evolution Equations,", Zürich Lectures in Advanced Mathematics, (2011).  doi: 10.4171/095.  Google Scholar

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.   Google Scholar

[24]

J. Shatah and M. Struwe, "Geometric Wave Equations,", Courant Lecture Notes, (1998).   Google Scholar

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