On self-similar blow up for the energy supercritical semilinear wave equation

Jihoi Kim

doi:10.5802/jep.282

On self-similar blow up for the energy supercritical semilinear wave equation
[Sur l’explosion auto-similaire pour l’équation d’onde semi-linéaire supercritique en énergie]

Jihoi Kim ¹

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1483-1542.

Résumé
Abstract

Nous analysons l’équation d’onde semi-linéaire supercritique en énergie

Φ_{t t} - Δ Φ - {| Φ |}^{p - 1} Φ = 0

dans l’espace $ℝ^{d}$ . Nous prouvons d’abord, dans un régime approprié de paramètres, l’existence d’une famille dénombrable de profils auto-similaires qui bifurquent à partir de la solution du soliton. Nous prouvons ensuite la stabilité non radiale en codimension finie de ces profils en adaptant le cadre fonctionnel de [22].

We analyse the energy supercritical semilinear wave equation

Φ_{t t} - Δ Φ - {| Φ |}^{p - 1} Φ = 0

in $ℝ^{d}$ space. We first prove in a suitable regime of parameters the existence of a countable family of self-similar profiles which bifurcate from the soliton solution. We then prove the non-radial finite codimensional stability of these profiles by adapting the functional setting of [22].

Reçu le : 2023-11-24
Accepté le : 2024-11-14
Publié le : 2024-11-28

DOI : 10.5802/jep.282

Classification : 35B44, 35C06, 35L05, 35L71
Keywords: Semi-linear wave equation, self-similar solution, blow up, focusing, energy super-critical, finite codimensional stability
Mots-clés : Équation d’onde semi-linéaire, solution auto-similaire, explosion, supercritique en énergie, stabilité en codimension finie

Affiliations des auteurs :

Jihoi Kim ¹

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2024__11__1483_0,
     author = {Jihoi Kim},
     title = {On self-similar blow up for the~energy~supercritical semilinear wave~equation},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1483--1542},
     publisher = {\'Ecole polytechnique},
     volume = {11},
     year = {2024},
     doi = {10.5802/jep.282},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.282/}
}

TY  - JOUR
AU  - Jihoi Kim
TI  - On self-similar blow up for the energy supercritical semilinear wave equation
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2024
SP  - 1483
EP  - 1542
VL  - 11
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.282/
DO  - 10.5802/jep.282
LA  - en
ID  - JEP_2024__11__1483_0
ER  -

%0 Journal Article
%A Jihoi Kim
%T On self-similar blow up for the energy supercritical semilinear wave equation
%J Journal de l’École polytechnique — Mathématiques
%D 2024
%P 1483-1542
%V 11
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.282/
%R 10.5802/jep.282
%G en
%F JEP_2024__11__1483_0

Jihoi Kim. On self-similar blow up for the energy supercritical semilinear wave equation. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1483-1542. doi : 10.5802/jep.282. https://jep.centre-mersenne.org/articles/10.5802/jep.282/

Bibliographie
Cité par

[1] Y. Bahri, Y. Martel & P. Raphaël - “Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations”, Ann. Henri Poincaré 22 (2021) no. 5, p. 1701-1749 | DOI | Zbl

[2] P. Biernat & P. Bizoń - “Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres”, Nonlinearity 24 (2011) no. 8, p. 2211-2228 | DOI | Zbl

[3] P. Bizoń, D. Maison & A. Wasserman - “Self-similar solutions of semilinear wave equations with a focusing nonlinearity”, Nonlinearity 20 (2007) no. 9, p. 2061-2074 | DOI | Zbl

[4] B. Cassano, L. Cossetti & L. Fanelli - “Improved Hardy-Rellich inequalities”, Comm. Pure Appl. Math. 21 (2022) no. 3, p. 867-889 | DOI | Zbl

[5] C. Collot - Type II blow up manifolds for the energy supercritical semilinear wave equation, Mem. Amer. Math. Soc., vol. 252, no. 1205, American Mathematical Society, Providence, RI, 2018 | DOI

[6] C. Collot, F. Merle & P. Raphaël - “Strongly anisotropic type II blow up at an isolated point”, J. Amer. Math. Soc. 33 (2020) no. 2, p. 527-607 | DOI | Zbl

[7] C. Collot, P. Raphaël & J. Szeftel - On the stability of type I blow up for the energy super critical heat equation, Mem. Amer. Math. Soc., vol. 260, no. 1255, American Mathematical Society, Providence, RI, 2019 | DOI

[8] W. Dai & T. Duyckaerts - “Self-similar solutions of focusing semi-linear wave equations in $ℝ^{N}$ ”, J. Evol. Equ. 21 (2021), p. 4703-4750 | DOI | Zbl

[9] R. Donninger & B. Schörkhuber - “On blowup in supercritical wave equations”, Comm. Math. Phys. 346 (2016), p. 907-943 | DOI | Zbl

[10] R. Duarte & J. D. Silva - “Weighted Gagliardo-Nirenberg interpolation inequalities”, J. Funct. Anal. 285 (2023) no. 5, article ID 110009, 49 pages | Zbl

[11] K.-J. Engel, R. Nagel & S. Brendle - One-parameter semigroups for linear evolution equations, Graduate Texts in Math., vol. 194, Springer, New York, 2000

[12] Y. Gao & J. Xue - “Local well-posedness and small data scattering for energy super-critical nonlinear wave equations”, Appl. Anal. 100 (2021) no. 3, p. 663-674 | DOI | Zbl

[13] I. Glogić & B. Schörkhuber - “Co-dimension one stable blowup for the supercritical cubic wave equation”, Adv. Math. 390 (2021), article ID 107930, 79 pages | DOI | Zbl

[14] M. A. Herrero & J. J. Velázquez - “A blow up result for semilinear heat equations in the supercritical case” (1992), preprint

[15] M. Hillairet & P. Raphaël - “Smooth type II blow-up solutions to the four-dimensional energy-critical wave equation”, Anal. PDE 5 (2012) no. 4, p. 777-829 | DOI | Zbl

[16] J. Krieger, W. Schlag & D. Tataru - “Renormalization and blow up for charge one equivariant critical wave maps”, Invent. Math. 171 (2008) no. 3, p. 543-615 | DOI | Zbl

[17] Y. Li - “Asymptotic behavior of positive solutions of equation $Δ u + K (x) u^{p} = 0$ in $ℝ^{n}$ ”, J. Differential Equations 95 (1992) no. 2, p. 304-330 | DOI | Zbl

[18] H. Matano & F. Merle - “On nonexistence of type II blowup for a supercritical nonlinear heat equation”, Comm. Pure Appl. Math. 57 (2004) no. 11, p. 1494-1541 | DOI | Zbl

[19] H. Matano & F. Merle - “Classification of type I and type II behaviors for a supercritical nonlinear heat equation”, J. Funct. Anal. 256 (2009) no. 4, p. 992-1064 | DOI | Zbl

[20] F. Merle, P. Raphaël & I. Rodnianski - “Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem”, Invent. Math. 193 (2013) no. 2, p. 249-365 | DOI | Zbl

[21] F. Merle, P. Raphaël & I. Rodnianski - “Type II blow up for the energy supercritical NLS”, Camb. J. Math. 3 (2015) no. 4, p. 439-617 | DOI | Zbl

[22] F. Merle, P. Raphaël, I. Rodnianski & J. Szeftel - “On blow up for the energy super critical defocusing nonlinear Schrödinger equations”, Invent. Math. 227 (2022) no. 1, p. 247-413 | DOI | Zbl

[23] F. Merle & H. Zaag - “Determination of the blow-up rate for the semilinear wave equation”, Amer. J. Math. 125 (2003) no. 5, p. 1147-1164 | DOI | Zbl

[24] F. Merle & H. Zaag - “Determination of the blow-up rate for a critical semilinear wave equation”, Math. Ann. 331 (2005) no. 2, p. 395-416 | DOI | Zbl

[25] F. Merle & H. Zaag - “Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension”, J. Funct. Anal. 253 (2007) no. 1, p. 43-121 | DOI | Zbl

[26] F. Merle & H. Zaag - “Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension”, in Sémin. Équ. Dériv. Partielles 2009–2010, École Polytechnique, Palaiseau, 2012, Exp. no. XI, 10 p.

[27] F. W. Olver - NIST handbook of mathematical functions, Cambridge University Press, Cambridge, 2010

[28] P. Raphaël & I. Rodnianski - “Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems”, Publ. Math. Inst. Hautes Études Sci. 115 (2012), p. 1-122 | DOI | Numdam | Zbl

[29] I. Rodnianski & J. Sterbenz - “On the formation of singularities in the critical $O (3)$ $σ$ -model”, Ann. of Math. (2) 172 (2010) no. 1, p. 187-242 | DOI | Zbl

[30] J. A. Toth & S. Zelditch - “Riemannian manifolds with uniformly bounded eigenfunctions”, Duke Math. J. 111 (2002) no. 1, p. 97-132 | DOI | Zbl

[31] G. Vainikko - “A smooth solution to a linear system of singular ODEs”, Z. Anal. Anwendungen 32 (2013) no. 3, p. 349-370 | DOI | Zbl

Cité par Sources :