Almost global solutions for non hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces

Jean-Marc Delort[1]; Jérémie Szeftel[2]

  • [1] Laboratoire Analyse Géométrie et Applications, UMR CNRS 7539 Institut Galilée, Université Paris-Nord, 99, Avenue J.-B. Clément, F-93430 Villetaneuse FRANCE
  • [2] Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA and Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex FRANCE

Journées Équations aux dérivées partielles (2005)

  • page 1-13
  • ISSN: 0752-0360

Abstract

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This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.

How to cite

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Delort, Jean-Marc, and Szeftel, Jérémie. "Almost global solutions for non hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces." Journées Équations aux dérivées partielles (2005): 1-13. <http://eudml.org/doc/10606>.

@article{Delort2005,
abstract = {This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.},
affiliation = {Laboratoire Analyse Géométrie et Applications, UMR CNRS 7539 Institut Galilée, Université Paris-Nord, 99, Avenue J.-B. Clément, F-93430 Villetaneuse FRANCE; Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA and Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex FRANCE},
author = {Delort, Jean-Marc, Szeftel, Jérémie},
journal = {Journées Équations aux dérivées partielles},
keywords = {almost global existence; nonlinear Klein-Gordon equation; revolution hypersurfaces; normal forms; small radial data},
language = {eng},
month = {6},
pages = {1-13},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Almost global solutions for non hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces},
url = {http://eudml.org/doc/10606},
year = {2005},
}

TY - JOUR
AU - Delort, Jean-Marc
AU - Szeftel, Jérémie
TI - Almost global solutions for non hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces
JO - Journées Équations aux dérivées partielles
DA - 2005/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 13
AB - This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.
LA - eng
KW - almost global existence; nonlinear Klein-Gordon equation; revolution hypersurfaces; normal forms; small radial data
UR - http://eudml.org/doc/10606
ER -

References

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  1. D. Bambusi: Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234 (2003), no. 2, 253–285. Zbl1032.37051MR1962462
  2. D. Bambusi and B. Grébert: Birkhoff normal form for PDEs with tame modulus, preprint (2004). Zbl1110.37057
  3. J. Bourgain: Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal. 6 (1996), no. 2, 201–230. Zbl0872.35007MR1384610
  4. J.-M. Delort: Existence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 1, 1–61 Zbl0990.35119MR1833089
  5. J.-M. Delort and J. Szeftel: Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not. (2004), no. 37, 1897–1966. Zbl1079.35070MR2056326
  6. J.-M. Delort and J. Szeftel: Almost orthogonality properties of products of eigenfunctions and applications to long-time existence for semi-linear Klein-Gordon equations on Zoll manifolds, preprint (2004). Zbl1079.35070MR2056326
  7. J.-M. Delort and J. Szeftel: Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces, preprint (2004). Zbl1115.35084
  8. S. Klainerman: The null condition and global existence to nonlinear wave equations, Lectures in Applied Mathematics 23, (1986), 293–326. Zbl0599.35105MR837683
  9. T. Ozawa, K. Tsutaya and Y. Tsutsumi: Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z, 222, (1996) 341–362. Zbl0877.35030MR1400196
  10. J. Shatah: Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38, (1985) 685–696. Zbl0597.35101MR803256

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