# American Institute of Mathematical Sciences

November  2013, 18(9): 2315-2329. doi: 10.3934/dcdsb.2013.18.2315

## 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

Received  March 2013 Revised  June 2013 Published  September 2013

We consider the semilinear wave equation in higher dimensions with power nonlinearity in the superconformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Vişan [22]. Our proof uses the similarity variables' setting. We consider the equation in that setting as a perturbation of the conformal case, and we handle the extra terms thanks to the ideas we already developed in [16] for perturbations of the pure power conformal case with lower order terms.
Citation: Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. Donninger, M. Huang, J. Krieger and W. Schlag, Exotic blowup solutions for the $u^5$ focusing wave equation in $\mathbbR^3$, (2012)., preprint, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [9] _________, Stable self-similar blow up for energy supercritical wave equations, (2012)., preprint, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] ________, Threshold solutions in the case of mass-shift for the critical klein-gordon equation, Trans. Amer. Math. Soc., (2013), to appear. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] R. Killip and M. Vişan, Smooth solutions to the nonlinear wave equation can blow up on Cantor sets, (2011)., arXiv:1103.5257v1., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. Donninger, M. Huang, J. Krieger and W. Schlag, Exotic blowup solutions for the $u^5$ focusing wave equation in $\mathbbR^3$, (2012)., preprint, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [9] _________, Stable self-similar blow up for energy supercritical wave equations, (2012)., preprint, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] ________, Threshold solutions in the case of mass-shift for the critical klein-gordon equation, Trans. Amer. Math. Soc., (2013), to appear. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] R. Killip and M. Vişan, Smooth solutions to the nonlinear wave equation can blow up on Cantor sets, (2011)., arXiv:1103.5257v1., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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