Stable blowup for wave equations in odd space dimensions

Donninger, Roland; Schörkhuber, Birgit

doi:10.1016/j.anihpc.2016.09.005

Donninger, Roland ; Schörkhuber, Birgit

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1181-1213.

Résumé

We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions $d \geq 5$ . We prove that for every $p > \frac{d + 3}{d - 1}$ there exists an open set of radial initial data in $H^{\frac{d + 1}{2}} \times H^{\frac{d - 1}{2}}$ 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.

MR Zbl

DOI : 10.1016/j.anihpc.2016.09.005

Mots clés : Nonlinear wave equations, Blowup, Stability

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     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/}
}

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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/

Bibliographie
Cité par

[1] Antonini, Christophe; Merle, Frank 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] Bizoń, Piotr; Chmaj, Tadeusz; Tabor, Zbisław On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, Volume 17 (2004) no. 6, pp. 2187–2201 | MR | Zbl

[3] Bizoń, Piotr; Maison, Dieter; Wasserman, Arthur Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity, Volume 20 (2007) no. 9, pp. 2061–2074 | MR | Zbl

[4] Bizoń, Poitr; Breitenlohner, Peter; Maison, Dieter; Wasserman, Arthur Self-similar solutions of the cubic wave equation, Nonlinearity, Volume 23 (2010) no. 2, pp. 225–236 | MR | Zbl

[5] Bulut, Aynur The defocusing energy-supercritical cubic nonlinear wave equation in dimension five, 2011 (preprint) | arXiv | Zbl

[6] Collot, Charles Type II blow up for the energy supercritical wave equation, 2014 (preprint) | arXiv | MR | Zbl

[7] Dodson, Benjamin; Lawrie, Andrew Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm, 2014 (preprint) | arXiv | MR | Zbl

[8] Donninger, Roland On stable self-similar blowup for equivariant wave maps, Commun. Pure Appl. Math., Volume 64 (2011) no. 8, pp. 1095–1147 | MR | Zbl

[9] Donninger, Roland; Krieger, Joachim Nonscattering solutions and blowup at infinity for the critical wave equation, Math. Ann., Volume 357 (2013) no. 1, pp. 89–163 | MR | Zbl

[10] Donninger, Roland; Schörkhuber, Birgit 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] Donninger, Roland; Schörkhuber, Birgit On blowup in supercritical wave equations, 2014 (preprint) | arXiv | MR

[12] Donninger, Roland; Schörkhuber, Birgit Stable blow up dynamics for energy supercritical wave equations, Trans. Am. Math. Soc., Volume 366 (2014) no. 4, pp. 2167–2189 | MR | Zbl

[13] Donninger, Roland; Schörkhuber, Birgit; Aichelburg, Peter C. 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] Duyckaerts, Thomas; Kenig, Carlos; Merle, Frank 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] Duyckaerts, Thomas; Kenig, Carlos; Merle, Frank Scattering for radial, bounded solutions of focusing supercritical wave equations, Int. Math. Res. Not. (2014) no. 1, pp. 224–258 | MR | Zbl

[16] Duyckaerts, Thomas; Kenig, Carlos; Merle, Frank Solutions of the focusing nonradial critical wave equation with the compactness property, 2014 (preprint) | arXiv | MR | Zbl

[17] Engel, Klaus-Jochen; Nagel, Rainer 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] Evans, Lawrence C. Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 2010 | MR | Zbl

[19] Hamza, Mohamed-Ali; Zaag, Hatem 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] Hillairet, Matthieu; Raphaël, Pierre 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] Jacek, Jendrej Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5, 2015 (preprint) | arXiv | MR | Zbl

[22] Kato, Tosio Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995 (Reprint of the 1980 edition) | MR | Zbl

[23] Kenig, Carlos E.; Merle, Frank 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] Killip, Rowan; Stovall, Betsy; Visan, Monica Blowup behaviour for the nonlinear Klein–Gordon equation, Math. Ann., Volume 358 (2014) no. 1–2, pp. 289–350 | MR | Zbl

[25] Killip, Rowan; Visan, Monica 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] Krieger, Joachim; Nahas, Joules Instability of type II blow up for the quintic nonlinear wave equation on $R^{3 + 1}$ , 2012 (preprint) | arXiv | Zbl

[27] Krieger, Joachim; Schlag, Wilhelm 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] Krieger, Joachim; Schlag, Wilhelm Large global solutions for energy supercritical nonlinear wave equations on $R^{3 + 1}$ , 2014 (preprint) | arXiv | MR | Zbl

[29] Krieger, Joachim; Schlag, Wilhelm; Tataru, Daniel Slow blow-up solutions for the $H^{1} (R^{3})$ critical focusing semilinear wave equation, Duke Math. J., Volume 147 (2009) no. 1, pp. 1–53 | MR | Zbl

[30] Krieger, Joachim; Wong, Willie 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] Levine, Howard A. Instability and nonexistence of global solutions to nonlinear wave equations of the form $P u_{t t} = - A u + F (u)$ , Trans. Am. Math. Soc., Volume 192 (1974), pp. 1–21 | MR | Zbl

[32] Lindblad, Hans; Sogge, Christopher D. 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] Merle, Frank; Raphaël, Pierre; Rodnianski, Igor Type II blow up for the energy supercritical NLS, 2014 (preprint) | arXiv | MR | Zbl

[34] Merle, Frank; Zaag, Hatem 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] Merle, Frank; Zaag, Hatem 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] Merle, Frank; Zaag, Hatem 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] Merle, Frank; Zaag, Hatem 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] Merle, Frank; Zaag, Hatem 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] Merle, Frank; Zaag, Hatem 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] Merle, Frank; Zaag, Hatem 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] Merle, Frank; Zaag, Hatem Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions, 2013 (preprint) | arXiv | MR | Zbl

[42] Merle, Frank; Zaag, Hatem 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] Olver, Frank W.J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. 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] Rauch, Jeffrey Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, vol. 133, American Mathematical Society, 2012 | MR | Zbl

[45] Tao, Terence 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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