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Blow-up results for semilinear wave equations in the superconformal case
1. | Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El-Manar, Campus Universitaire 1060, Tunis, Tunisia |
2. | Université Paris 13, Sorbonne Paris Cit, LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France |
References:
[1] |
C. Antonini and F. Merle, Optimal bounds on positive blow-up solutions for a semilinear wave equation, Internat. Math. Res. Notices, (2001), 1141-1167.
doi: 10.1155/S107379280100054X. |
[2] |
P. Bizoń, P. Breitenlohner, D. Maison and A. Wasserman, Self-similar solutions of the cubic wave equation, Nonlinearity, 23 (2010), 225-236.
doi: 10.1088/0951-7715/23/2/002. |
[3] |
P. Bizoń, T. Chmaj and N. Szpak, Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation, J. Math. Phys., 52 (2011), 103703, 11.
doi: 10.1063/1.3645363. |
[4] |
P. Bizoń, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 17 (2004), 2187-2201.
doi: 10.1088/0951-7715/17/6/009. |
[5] |
R. Côte and H. Zaag, Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math., 66 (2013), 1541-1581.
doi: 10.1002/cpa.21452. |
[6] |
R. Donninger, M. Huang, J. Krieger and W. Schlag, Exotic blowup solutions for the $u^5$ focusing wave equation in $\mathbb{R}^3$, (2012). preprint, arXiv:1212.4718. |
[7] |
R. Donninger and W. Schlag, Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation, Nonlinearity, 24 (2011), 2547-2562.
doi: 10.1088/0951-7715/24/9/009. |
[8] |
R. Donninger and B. Schörkhuber, Stable self-similar blow up for energy subcritical wave equations, Dyn. Partial Differ. Equ., 9 (2012), 63-87.
doi: 10.4310/DPDE.2012.v9.n1.a3. |
[9] |
_________, Stable self-similar blow up for energy supercritical wave equations, (2012). preprint, arXiv:1207.7046. |
[10] |
T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type {II blow-up solutions of the energy-critical wave equation}, J. Eur. Math. Soc. (JEMS), 13 (2011), 533-599.
doi: 10.4171/JEMS/261. |
[11] |
________, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge J. Math, 1 (2013), 75-144.
doi: 10.4310/CJM.2013.v1.n1.a3. |
[12] |
________, Profiles of bounded radial solutions of the focusing, energy-critical wave equation, Geom. Funct. Anal., 22 (2012), 639-698.
doi: 10.1007/s00039-012-0174-7. |
[13] |
________, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS), 14 (2012), 1389-1454.
doi: 10.4171/JEMS/336. |
[14] |
T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP, (2008), pp. Art ID rpn002, 67.
doi: 10.1093/imrp/rpn002. |
[15] |
M. Hamza and H. Zaag, Blow-up behavior for the Klein-Gordon and other perturbed semilinear wave equations, Bull. Sci. Math., (2013), to appear.
doi: 10.1016/j.bulsci.2013.05.004. |
[16] |
________, A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case, J. Hyperbolic Differ. Equ., 9 (2012), 195-221.
doi: 10.1142/S0219891612500063. |
[17] |
________, A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations, Nonlinearity, 25 (2012), 2759-2773.
doi: 10.1088/0951-7715/25/9/2759. |
[18] |
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. |
[19] |
________, Threshold solutions in the case of mass-shift for the critical klein-gordon equation, Trans. Amer. Math. Soc., (2013), to appear. |
[20] |
C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Mathematica, 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[21] |
________, Radial solutions to energy supercritical wave equations in odd dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 1365-1381.
doi: 10.3934/dcds.2011.31.1365. |
[22] |
R. Killip, B. Stovall, and M. Vişan, Blowup behaviour for the nonlinear Klein-Gordon equation, Math. Ann., (2013), to appear.
doi: 10.1007/s00208-013-0960-z. |
[23] |
R. Killip and M. Vişan, Smooth solutions to the nonlinear wave equation can blow up on Cantor sets, (2011). arXiv:1103.5257v1. |
[24] |
J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, Amer. J. Math., 135 (2013), 935-965.
doi: 10.1353/ajm.2013.0034. |
[25] |
________, Global dynamics of the nonradial energy-critical wave equation above the ground state energy, Discrete Contin. Dyn. Syst. 33 (2013), 2423-2450.
doi: 10.3934/dcds.2013.33.2423. |
[26] |
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. |
[27] |
H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $P u_{t t} = -A u + \mathcal F (u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. |
[28] |
F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147-1164.
doi: 10.1353/ajm.2003.0033. |
[29] |
________, Blow-up rate near the blow-up surface for semilinear wave equations, Internat. Math. Res. Notices, (2005), 1127-1156.
doi: 10.1155/IMRN.2005.1127. |
[30] |
________, Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann., 331 (2005), 395-416.
doi: 10.1007/s00208-004-0587-1. |
[31] |
________, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., 253 (2007), 43-121.
doi: 10.1016/j.jfa.2007.03.007. |
[32] |
________, Openness of the set of non characteristic points and regularity of the blow-up curve for the 1 d semilinear wave equation, Comm. Math. Phys., 282 (2008), 55-86.
doi: 10.1007/s00220-008-0532-3. |
[33] |
________, Isolatedness of characteristic points for a semilinear wave equation in one space dimension, in Séminaire sur les Équations aux Dérivées Partielles, 2009-2010, École Polytech., Palaiseau, 2010, Exp. No. 11, 10p. |
[34] |
________, Blow-up behavior outside the origin for a semilinear wave equation in the radial case, Bull. Sci. Math., 135 (2011), 353-373.
doi: 10.1016/j.bulsci.2011.03.001. |
[35] |
________, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math., 134 (2012), 581-648.
doi: 10.1353/ajm.2012.0021. |
[36] |
________, Isolatedness of characteristic points for a semilinear wave equation in one space dimension, Duke Math. J., 161 (2012), 2837-2908.
doi: 10.1215/00127094-1902040. |
[37] |
K. Nakanishi and W. Schlag, "Invariant Manifolds and Dispersive Hamiltonian Evolution Equations," Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2011.
doi: 10.4171/095. |
[38] |
G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original, A Wiley-Interscience Publication.
doi: 10.1002/9781118032954. |
show all references
References:
[1] |
C. Antonini and F. Merle, Optimal bounds on positive blow-up solutions for a semilinear wave equation, Internat. Math. Res. Notices, (2001), 1141-1167.
doi: 10.1155/S107379280100054X. |
[2] |
P. Bizoń, P. Breitenlohner, D. Maison and A. Wasserman, Self-similar solutions of the cubic wave equation, Nonlinearity, 23 (2010), 225-236.
doi: 10.1088/0951-7715/23/2/002. |
[3] |
P. Bizoń, T. Chmaj and N. Szpak, Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation, J. Math. Phys., 52 (2011), 103703, 11.
doi: 10.1063/1.3645363. |
[4] |
P. Bizoń, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 17 (2004), 2187-2201.
doi: 10.1088/0951-7715/17/6/009. |
[5] |
R. Côte and H. Zaag, Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension, Comm. Pure Appl. Math., 66 (2013), 1541-1581.
doi: 10.1002/cpa.21452. |
[6] |
R. Donninger, M. Huang, J. Krieger and W. Schlag, Exotic blowup solutions for the $u^5$ focusing wave equation in $\mathbb{R}^3$, (2012). preprint, arXiv:1212.4718. |
[7] |
R. Donninger and W. Schlag, Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation, Nonlinearity, 24 (2011), 2547-2562.
doi: 10.1088/0951-7715/24/9/009. |
[8] |
R. Donninger and B. Schörkhuber, Stable self-similar blow up for energy subcritical wave equations, Dyn. Partial Differ. Equ., 9 (2012), 63-87.
doi: 10.4310/DPDE.2012.v9.n1.a3. |
[9] |
_________, Stable self-similar blow up for energy supercritical wave equations, (2012). preprint, arXiv:1207.7046. |
[10] |
T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type {II blow-up solutions of the energy-critical wave equation}, J. Eur. Math. Soc. (JEMS), 13 (2011), 533-599.
doi: 10.4171/JEMS/261. |
[11] |
________, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge J. Math, 1 (2013), 75-144.
doi: 10.4310/CJM.2013.v1.n1.a3. |
[12] |
________, Profiles of bounded radial solutions of the focusing, energy-critical wave equation, Geom. Funct. Anal., 22 (2012), 639-698.
doi: 10.1007/s00039-012-0174-7. |
[13] |
________, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS), 14 (2012), 1389-1454.
doi: 10.4171/JEMS/336. |
[14] |
T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP, (2008), pp. Art ID rpn002, 67.
doi: 10.1093/imrp/rpn002. |
[15] |
M. Hamza and H. Zaag, Blow-up behavior for the Klein-Gordon and other perturbed semilinear wave equations, Bull. Sci. Math., (2013), to appear.
doi: 10.1016/j.bulsci.2013.05.004. |
[16] |
________, A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case, J. Hyperbolic Differ. Equ., 9 (2012), 195-221.
doi: 10.1142/S0219891612500063. |
[17] |
________, A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations, Nonlinearity, 25 (2012), 2759-2773.
doi: 10.1088/0951-7715/25/9/2759. |
[18] |
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. |
[19] |
________, Threshold solutions in the case of mass-shift for the critical klein-gordon equation, Trans. Amer. Math. Soc., (2013), to appear. |
[20] |
C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Mathematica, 201 (2008), 147-212.
doi: 10.1007/s11511-008-0031-6. |
[21] |
________, Radial solutions to energy supercritical wave equations in odd dimensions, Discrete Contin. Dyn. Syst., 31 (2011), 1365-1381.
doi: 10.3934/dcds.2011.31.1365. |
[22] |
R. Killip, B. Stovall, and M. Vişan, Blowup behaviour for the nonlinear Klein-Gordon equation, Math. Ann., (2013), to appear.
doi: 10.1007/s00208-013-0960-z. |
[23] |
R. Killip and M. Vişan, Smooth solutions to the nonlinear wave equation can blow up on Cantor sets, (2011). arXiv:1103.5257v1. |
[24] |
J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, Amer. J. Math., 135 (2013), 935-965.
doi: 10.1353/ajm.2013.0034. |
[25] |
________, Global dynamics of the nonradial energy-critical wave equation above the ground state energy, Discrete Contin. Dyn. Syst. 33 (2013), 2423-2450.
doi: 10.3934/dcds.2013.33.2423. |
[26] |
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. |
[27] |
H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $P u_{t t} = -A u + \mathcal F (u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. |
[28] |
F. Merle and H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math., 125 (2003), 1147-1164.
doi: 10.1353/ajm.2003.0033. |
[29] |
________, Blow-up rate near the blow-up surface for semilinear wave equations, Internat. Math. Res. Notices, (2005), 1127-1156.
doi: 10.1155/IMRN.2005.1127. |
[30] |
________, Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann., 331 (2005), 395-416.
doi: 10.1007/s00208-004-0587-1. |
[31] |
________, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., 253 (2007), 43-121.
doi: 10.1016/j.jfa.2007.03.007. |
[32] |
________, Openness of the set of non characteristic points and regularity of the blow-up curve for the 1 d semilinear wave equation, Comm. Math. Phys., 282 (2008), 55-86.
doi: 10.1007/s00220-008-0532-3. |
[33] |
________, Isolatedness of characteristic points for a semilinear wave equation in one space dimension, in Séminaire sur les Équations aux Dérivées Partielles, 2009-2010, École Polytech., Palaiseau, 2010, Exp. No. 11, 10p. |
[34] |
________, Blow-up behavior outside the origin for a semilinear wave equation in the radial case, Bull. Sci. Math., 135 (2011), 353-373.
doi: 10.1016/j.bulsci.2011.03.001. |
[35] |
________, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math., 134 (2012), 581-648.
doi: 10.1353/ajm.2012.0021. |
[36] |
________, Isolatedness of characteristic points for a semilinear wave equation in one space dimension, Duke Math. J., 161 (2012), 2837-2908.
doi: 10.1215/00127094-1902040. |
[37] |
K. Nakanishi and W. Schlag, "Invariant Manifolds and Dispersive Hamiltonian Evolution Equations," Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2011.
doi: 10.4171/095. |
[38] |
G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original, A Wiley-Interscience Publication.
doi: 10.1002/9781118032954. |
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