Comptes Rendus
no. 3
Partial Differential Equations
Ill-posedness of H1-supercritical waves
[Instabilité des ondes H1-surcritiques]

Présenté par : Gilles Lebeau

Slim Ibrahim1 ; Mohamed Majdoub2 ; Nader Masmoudi3
1 Department of Mathematics & Statistics, Arizona State University, Tempe, AZ 85287-1804, USA
2 Département de mathématiques, faculté des sciences de Tunis, campus universitaire 2092, Tunis, Tunisia
3 The Courant Institute of Mathematical Sciences, NY University, 251 Mercer St., New York, NY 10012, USA
Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 133-138.

Nous étendons les résultats de Lebeau (2005) et Christ et al. (2007) aux cas de nonlinearités surcritiques quelconques. On traite aussi le cas 2D pour des nonlinéarités à croissance exponentielle. La preuve utilise la vitesse finie de propagation et une étude quantitative de l'E.D.O. associée.

We extend the results of Lebeau (2005) and Christ et al. (2007) to more general H1-supercritical nonlinearities. We also extend those results to the 2D case for exponentially growing nonlinearities. The proof uses the finite speed of propagation and a quantitative study of the associated O.D.E. It does not require any scaling invariance of the equation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.06.008

Slim Ibrahim 1 ; Mohamed Majdoub 2 ; Nader Masmoudi 3

1 Department of Mathematics & Statistics, Arizona State University, Tempe, AZ 85287-1804, USA
2 Département de mathématiques, faculté des sciences de Tunis, campus universitaire 2092, Tunis, Tunisia
3 The Courant Institute of Mathematical Sciences, NY University, 251 Mercer St., New York, NY 10012, USA 
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Slim Ibrahim; Mohamed Majdoub; Nader Masmoudi. Ill-posedness of $ {H}^{1}$-supercritical waves. Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 133-138. doi : 10.1016/j.crma.2007.06.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.008/

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  • Dana Mendelson Symplectic non-squeezing for the cubic nonlinear Klein–Gordon equation on T3, Journal of Functional Analysis, Volume 272 (2017) no. 7, p. 3019 | DOI:10.1016/j.jfa.2016.12.025
  • Antoine Choffrut; Oana Pocovnicu Ill-Posedness of the Cubic Nonlinear Half-Wave Equation and Other Fractional NLS on the Real Line, International Mathematics Research Notices (2016), p. rnw246 | DOI:10.1093/imrn/rnw246
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  • Tarek Saanouni A note on global well-posedness and blow-up of some semilinear evolution equations, Evolution Equations Control Theory, Volume 4 (2015) no. 3, p. 355 | DOI:10.3934/eect.2015.4.355
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  • M. Majdoub; N. Masmoudi On Uniqueness for Supercritical Nonlinear Wave and Schrodinger Equations, International Mathematics Research Notices (2014) | DOI:10.1093/imrn/rnu002
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  • Slim Ibrahim; Mohamed Majdoub; Nader Masmoudi Well- and ill-posedness issues for energy supercritical waves, Analysis PDE, Volume 4 (2011) no. 2, p. 341 | DOI:10.2140/apde.2011.4.341
  • Hajer Bahouri; Mohamed Majdoub; Nader Masmoudi On the lack of compactness in the 2D critical Sobolev embedding, Journal of Functional Analysis, Volume 260 (2011) no. 1, p. 208 | DOI:10.1016/j.jfa.2010.08.016
  • Slim Ibrahim; Mohamed Majdoub; Nader Masmoudi; Kenji Nakanishi Scattering for the two-dimensional energy-critical wave equation, Duke Mathematical Journal, Volume 150 (2009) no. 2 | DOI:10.1215/00127094-2009-053
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