Ill-posedness of $ {H}^{1}$-supercritical waves

Slim Ibrahim; Mohamed Majdoub; Nader Masmoudi

doi:10.1016/j.crma.2007.06.008

Partial Differential Equations

Ill-posedness of

H^{1}

-supercritical waves

Presented by: Gilles Lebeau

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

Comptes Rendus. Mathématique, Volume 345 (2007) no. 3, pp. 133-138.

Abstract
Résumé

We extend the results of Lebeau (2005) and Christ et al. (2007) to more general $H^{1}$ -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.

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.

Received: 2007-05-09
Accepted: 2007-06-05
Published online: 2007-07-20

DOI: 10.1016/j.crma.2007.06.008

Author's affiliations:

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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     author = {Slim Ibrahim and Mohamed Majdoub and Nader Masmoudi},
     title = {Ill-posedness of $ {H}^{1}$-supercritical waves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {133--138},
     publisher = {Elsevier},
     volume = {345},
     number = {3},
     year = {2007},
     doi = {10.1016/j.crma.2007.06.008},
     language = {en},
}

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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/

References
Cited by

[1] T. Alazard, R. Carles, Sequential loss of regularity for super-critical nonlinear Schrödinger equations, Preprint

[2] N. Burq, P. Gérard, S. Ibrahim, Instability results for nonlinear Schrödinger and wave equations, Preprint

[3] N. Burq; P. Gérard; N. Tzvetkov Bilinear estimates and the nonlinear Schödinger equation on surfaces, Inven. Math., Volume 159 (2005), pp. 187-223

[4] M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, | arXiv

[5] J. Ginibre; G. Velo The global Cauchy problem for nonlinear Klein–Gordon equation, Math. Z, Volume 189 (1985), pp. 487-505

[6] J. Ginibre; A. Soffer; G. Velo The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., Volume 110 (1992), pp. 96-130

[7] S. Ibrahim; M. Majdoub; N. Masmoudi Global solutions for a semilinear 2D Klein–Gordon equation with exponential type nonlinearity, Commun. Pure Appl. Math., Volume 59 (2006) no. 11, pp. 1639-1658

[8] S. Ibrahim, M. Majdoub, N. Masmoudi, On the well-posedness of some NLW and NLS equations, in preparation

[9] C. Kenig; G. Ponce; L. Vega On the ill-posedness of some canonical dispersive equations, Duke Math. J., Volume 106 (2001) no. 3, pp. 617-633

[10] G. Lebeau Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège, Volume 70 (2001) no. 4–6, pp. 267-306

[11] G. Lebeau Perte de régularuté pour l'équation des ondes surcritique, Bull. Soc. Math. France, Volume 133 (2005), pp. 145-157

[12] J. Moser A sharp form of an inequality of N. Trudinger, Indiana Univ. Math. J., Volume 20 (1971), pp. 1077-1092

[13] M. Nakamura; T. Ozawa Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z., Volume 231 (1999), pp. 479-487

[14] M. Nakamura; T. Ozawa The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order, Discrete and Continuous Dynamical Systems, Volume 5 (1999) no. 1, pp. 215-231

[15] J. Shatah; M. Struwe Well-posedness in the energy space for semilinear wave equation with critical growth, IMRN, Volume 7 (1994), pp. 303-309

[16] W. Strauss Nonlinear Wave Equations, Conf. Board of the Math. Sciences, vol. 73, Amer. Math. Soc., 1989

[17] M. Struwe Semilinear wave equations, Bull. Amer. Math. Soc. (N.S.), Volume 26 (1992), pp. 53-85

[18] T. Tao Global regularity for a logarithmically supercritical defocussing nonlinear wave equation for spherically symmetric data, J. Hyperbolic Diff. Eq., Volume 4 (2007), pp. 259-266

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