Blow-up results for semilinear wave equations in the superconformal case

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

Blow-up results for semilinear wave equations in the superconformal case

Mohamed-Ali Hamza ^1, and Hatem Zaag ^2,

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

Abstract
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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.

Keywords: perturbations., Wave equation, blow-up.

Mathematics Subject Classification: 35L05, 35B20, 35B4.

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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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). 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). 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. 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. 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. 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). 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. 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. 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). 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. 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), 33 (2013), 2423. 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. 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. 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. 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. 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. 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. 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. 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, (2010), 2009. Google Scholar
[34]	________, Blow-up behavior outside the origin for a semilinear wave equation in the radial case,, Bull. Sci. Math., 135 (2011), 353. 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. 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. 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, (2011). doi: 10.4171/095. Google Scholar
[38]	G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics (New York), (1999). doi: 10.1002/9781118032954. Google Scholar

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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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). 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). 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. 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. 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. 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). 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. 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. 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). 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. 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), 33 (2013), 2423. 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. 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. 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. 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. 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. 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. 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. 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, (2010), 2009. Google Scholar
[34]	________, Blow-up behavior outside the origin for a semilinear wave equation in the radial case,, Bull. Sci. Math., 135 (2011), 353. 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. 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. 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, (2011). doi: 10.4171/095. Google Scholar
[38]	G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics (New York), (1999). doi: 10.1002/9781118032954. Google Scholar

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