# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 131-175. [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), 271-297. doi: 10.1002/cpa.3160420304. [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), 533-599. doi: 10.4171/JEMS/261. [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. [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. [6] T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, preprint, (). [7] T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x. [8] T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP, (2008). [9] J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical non-linear wave equation, J. Funct. Anal., 110 (1992), 96-130. doi: 10.1016/0022-1236(92)90044-J. [10] S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. doi: 10.2140/apde.2011.4.405. [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), 645-675. doi: 10.1007/s00222-006-0011-4. [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), 147-212. doi: 10.1007/s11511-008-0031-6. [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., (). [14] J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation, Amer. J. Math., 129 (2007), 843-913. doi: 10.1353/ajm.2007.0021. [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), 1-53. doi: 10.1215/00127094-2009-005. [16] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6. [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), 399-425. doi: 10.1155/S1073792898000270. [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. [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), 2299-2233. doi: 10.1016/j.jde.2010.10.027. [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), 1-45. doi: 10.1007/s00526-011-0424-9. [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), 809-851. doi: 10.1007/s00205-011-0462-7. [22] K. Nakanishi and W. Schlag, "Invariant Manifolds and Dispersive Hamiltonian Evolution Equations," Zürich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095. [23] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. [24] J. Shatah and M. Struwe, "Geometric Wave Equations," Courant Lecture Notes, AMS, 1998.

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##### References:
 [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. [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), 271-297. doi: 10.1002/cpa.3160420304. [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), 533-599. doi: 10.4171/JEMS/261. [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. [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. [6] T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation,, preprint, (). [7] T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x. [8] T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP, (2008). [9] J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical non-linear wave equation, J. Funct. Anal., 110 (1992), 96-130. doi: 10.1016/0022-1236(92)90044-J. [10] S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. doi: 10.2140/apde.2011.4.405. [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), 645-675. doi: 10.1007/s00222-006-0011-4. [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), 147-212. doi: 10.1007/s11511-008-0031-6. [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., (). [14] J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation, Amer. J. Math., 129 (2007), 843-913. doi: 10.1353/ajm.2007.0021. [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), 1-53. doi: 10.1215/00127094-2009-005. [16] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6. [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), 399-425. doi: 10.1155/S1073792898000270. [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. [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), 2299-2233. doi: 10.1016/j.jde.2010.10.027. [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), 1-45. doi: 10.1007/s00526-011-0424-9. [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), 809-851. doi: 10.1007/s00205-011-0462-7. [22] K. Nakanishi and W. Schlag, "Invariant Manifolds and Dispersive Hamiltonian Evolution Equations," Zürich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095. [23] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. [24] J. Shatah and M. Struwe, "Geometric Wave Equations," Courant Lecture Notes, AMS, 1998.
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