We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions . We prove that for every there exists an open set of radial initial data in such that the corresponding solution exists in a backward lightcone and approaches the ODE blowup profile. The result covers the entire range of energy supercritical nonlinearities and extends our previous work for the three-dimensional radial wave equation to higher space dimensions.
@article{AIHPC_2017__34_5_1181_0, author = {Donninger, Roland and Sch\"orkhuber, Birgit}, title = {Stable blowup for wave equations in odd space dimensions}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1181--1213}, publisher = {Elsevier}, volume = {34}, number = {5}, year = {2017}, doi = {10.1016/j.anihpc.2016.09.005}, zbl = {1395.35041}, mrnumber = {3742520}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.09.005/} }
TY - JOUR AU - Donninger, Roland AU - Schörkhuber, Birgit TI - Stable blowup for wave equations in odd space dimensions JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 1181 EP - 1213 VL - 34 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2016.09.005/ DO - 10.1016/j.anihpc.2016.09.005 LA - en ID - AIHPC_2017__34_5_1181_0 ER -
%0 Journal Article %A Donninger, Roland %A Schörkhuber, Birgit %T Stable blowup for wave equations in odd space dimensions %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 1181-1213 %V 34 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2016.09.005/ %R 10.1016/j.anihpc.2016.09.005 %G en %F AIHPC_2017__34_5_1181_0
Donninger, Roland; Schörkhuber, Birgit. Stable blowup for wave equations in odd space dimensions. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1181-1213. doi : 10.1016/j.anihpc.2016.09.005. http://www.numdam.org/articles/10.1016/j.anihpc.2016.09.005/
[1] Optimal bounds on positive blow-up solutions for a semilinear wave equation, Int. Math. Res. Not. (2001) no. 21, pp. 1141–1167 | MR | Zbl
[2] On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, Volume 17 (2004) no. 6, pp. 2187–2201 | MR | Zbl
[3] Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity, Volume 20 (2007) no. 9, pp. 2061–2074 | MR | Zbl
[4] Self-similar solutions of the cubic wave equation, Nonlinearity, Volume 23 (2010) no. 2, pp. 225–236 | MR | Zbl
[5] The defocusing energy-supercritical cubic nonlinear wave equation in dimension five, 2011 (preprint) | arXiv | Zbl
[6] Type II blow up for the energy supercritical wave equation, 2014 (preprint) | arXiv | MR | Zbl
[7] Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm, 2014 (preprint) | arXiv | MR | Zbl
[8] On stable self-similar blowup for equivariant wave maps, Commun. Pure Appl. Math., Volume 64 (2011) no. 8, pp. 1095–1147 | MR | Zbl
[9] Nonscattering solutions and blowup at infinity for the critical wave equation, Math. Ann., Volume 357 (2013) no. 1, pp. 89–163 | MR | Zbl
[10] Stable self-similar blow up for energy subcritical wave equations, Dyn. Partial Differ. Equ., Volume 9 (2012) no. 1, pp. 63–87 | MR | Zbl
[11] On blowup in supercritical wave equations, 2014 (preprint) | arXiv | MR
[12] Stable blow up dynamics for energy supercritical wave equations, Trans. Am. Math. Soc., Volume 366 (2014) no. 4, pp. 2167–2189 | MR | Zbl
[13] On stable self-similar blow up for equivariant wave maps: the linearized problem, Ann. Henri Poincaré, Volume 13 (2012) no. 1, pp. 103–144 | MR | Zbl
[14] Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math., Volume 1 (2013) no. 1, pp. 75–144 | MR | Zbl
[15] Scattering for radial, bounded solutions of focusing supercritical wave equations, Int. Math. Res. Not. (2014) no. 1, pp. 224–258 | MR | Zbl
[16] Solutions of the focusing nonradial critical wave equation with the compactness property, 2014 (preprint) | arXiv | MR | Zbl
[17] One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000 (With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt) | MR | Zbl
[18] Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 2010 | MR | Zbl
[19] Blow-up results for semilinear wave equations in the superconformal case, Discrete Contin. Dyn. Syst., Ser. B, Volume 18 (2013) no. 9, pp. 2315–2329 | MR | Zbl
[20] Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation, Anal. PDE, Volume 5 (2012) no. 4, pp. 777–829 | MR | Zbl
[21] Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5, 2015 (preprint) | arXiv | MR | Zbl
[22] Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995 (Reprint of the 1980 edition) | MR | Zbl
[23] Radial solutions to energy supercritical wave equations in odd dimensions, Discrete Contin. Dyn. Syst., Volume 31 (2011) no. 4, pp. 1365–1381 | MR | Zbl
[24] Blowup behaviour for the nonlinear Klein–Gordon equation, Math. Ann., Volume 358 (2014) no. 1–2, pp. 289–350 | MR | Zbl
[25] The defocusing energy-supercritical nonlinear wave equation in three space dimensions, Trans. Am. Math. Soc., Volume 363 (2011) no. 7, pp. 3893–3934 | MR | Zbl
[26] Instability of type II blow up for the quintic nonlinear wave equation on , 2012 (preprint) | arXiv | Zbl
[27] Full range of blow up exponents for the quintic wave equation in three dimensions, J. Math. Pures Appl. (9), Volume 101 (2014) no. 6, pp. 873–900 | MR | Zbl
[28] Large global solutions for energy supercritical nonlinear wave equations on , 2014 (preprint) | arXiv | MR | Zbl
[29] Slow blow-up solutions for the critical focusing semilinear wave equation, Duke Math. J., Volume 147 (2009) no. 1, pp. 1–53 | MR | Zbl
[30] On type I blow-up formation for the critical NLW, Commun. Partial Differ. Equ., Volume 39 (2014) no. 9, pp. 1718–1728 | MR | Zbl
[31] Instability and nonexistence of global solutions to nonlinear wave equations of the form , Trans. Am. Math. Soc., Volume 192 (1974), pp. 1–21 | MR | Zbl
[32] On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., Volume 130 (1995) no. 2, pp. 357–426 | MR | Zbl
[33] Type II blow up for the energy supercritical NLS, 2014 (preprint) | arXiv | MR | Zbl
[34] Determination of the blow-up rate for the semilinear wave equation, Am. J. Math., Volume 125 (2003) no. 5, pp. 1147–1164 | MR | Zbl
[35] Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann., Volume 331 (2005) no. 2, pp. 395–416 | MR | Zbl
[36] Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., Volume 253 (2007) no. 1, pp. 43–121 | MR | Zbl
[37] Openness of the set of non-characteristic points and regularity of the blow-up curve for the 1 D semilinear wave equation, Commun. Math. Phys., Volume 282 (2008) no. 1, pp. 55–86 | MR | Zbl
[38] Blow-up behavior outside the origin for a semilinear wave equation in the radial case, Bull. Sci. Math., Volume 135 (2011) no. 4, pp. 353–373 | MR | Zbl
[39] Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Am. J. Math., Volume 134 (2012) no. 3, pp. 581–648 | MR | Zbl
[40] Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension, Duke Math. J., Volume 161 (2012) no. 15, pp. 2837–2908 | MR | Zbl
[41] Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions, 2013 (preprint) | arXiv | MR | Zbl
[42] On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Commun. Math. Phys., Volume 333 (2015) no. 3, pp. 1529–1562 | MR | Zbl
[43] NIST Handbook of Mathematical Functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, 2010 With 1 CD-ROM (Windows, Macintosh and UNIX) | MR | Zbl
[44] Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, vol. 133, American Mathematical Society, 2012 | MR | Zbl
[45] Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 106, American Mathematical Society, 2006 | MR | Zbl
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