Comptes Rendus
no. 1-2
Partial Differential Equations/Mathematical Physics
On wave propagation in the Anti-de Sitter cosmology
[Sur la propagation des ondes dans la cosmologie d'Anti-de Sitter]

Présenté par : Yvonne Choquet-Bruhat

Alain Bachelot1
1 Université de Bordeaux, institut de mathématiques, UMR CNRS 5251, 33405 Talence cedex, France
Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 47-51.

Nous étudions l'équation de Klein–Gordon sur la variété de Poincaré Anti-de Sitter de dimension 5. En dépit de la perte d'hyperbolicité globale, le problème de Cauchy est bien posé dans un cadre d'énergie finie. On établit plusieurs résultats de comportements asymptotiques. Nous considérons aussi le modèle cosmologique de la membrane de Minkowski de tension négative, pour laquelle nous résolvons le problème mixte.

We investigate the Klein–Gordon equation on the Poincaré patch of the 5-dimensional Anti-de Sitter universe. Despite the loss of the global hyperbolicity, the Cauchy problem is well-posed in the space of the finite energy data. We establish several results of asymptotic behaviours. We consider also the cosmological model of the Minkowski brane with a negative tension, for which we solve the mixed problem.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.11.033
Alain Bachelot 1

1 Université de Bordeaux, institut de mathématiques, UMR CNRS 5251, 33405 Talence cedex, France 
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Alain Bachelot. On wave propagation in the Anti-de Sitter cosmology. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 47-51. doi : 10.1016/j.crma.2010.11.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.033/

[1] A. Bachelot The Dirac system on the Anti-de Sitter universe, Commun. Math. Phys., Volume 283 (2008), pp. 127-167

[2] A. Bachelot Wave propagation and scattering for the RS2 brane cosmology model, JHDE, Volume 6 (2009) no. 4, pp. 809-861

[3] A. Bachelot The Klein–Gordon equation in Anti-de Sitter cosmology, 2010 | arXiv

[4] Y. Choquet-Bruhat; D. Christodoulou Elliptic systems in Hs,δ spaces on manifolds which are Euclidean at infinity, Acta Math., Volume 146 (1981), pp. 129-150

[5] P. d'Ancona; V. Georgiev; H. Kubo Weighted decay estimates for the wave equation, J. Differential Equations, Volume 177 (2001), pp. 146-2008

[6] A. Ishibashi; R.M. Wald Dynamics in non-globally-hyperbolic, static space–times: III. Anti-de-Sitter space–time, Class. Quantum Grav., Volume 21 (2004), pp. 2981-3013

[7] Ph.D. Mannheim Brane-Localized Gravity, World Scientific, 2005

[8] A. Vasy The wave equation on asymptotically Anti-de Sitter spaces, 2009 | arXiv

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