Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces

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

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

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 5, page 1419-1456
  • ISSN: 0373-0956

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. "Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces." Annales de l’institut Fourier 56.5 (2006): 1419-1456. <http://eudml.org/doc/10181>.

@article{Delort2006,
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 = {Université Paris-Nord, Institut Galilée UMR CNRS 7539 Laboratoire Analyse Géométrie et Applications 99, Avenue J.-B. Clément 93430 Villetaneuse (France); Princeton University Department of Mathematics Fine Hall, Washington Road Princeton NJ 08544-1000 (USA) and Université Bordeaux 1, UMR CNRS 5466 Mathématiques Appliquées de Bordeaux 351 cours de la Libération 33405 Talence cedex (France)},
author = {Delort, Jean-Marc, Szeftel, Jérémie},
journal = {Annales de l’institut Fourier},
keywords = {Almost global solutions; nonlinear Klein-Gordon equation; radial hypersurfaces; small radial data},
language = {eng},
number = {5},
pages = {1419-1456},
publisher = {Association des Annales de l’institut Fourier},
title = {Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces},
url = {http://eudml.org/doc/10181},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Delort, Jean-Marc
AU - Szeftel, Jérémie
TI - Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1419
EP - 1456
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 solutions; nonlinear Klein-Gordon equation; radial hypersurfaces; small radial data
UR - http://eudml.org/doc/10181
ER -

References

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  2. D. Bambusi, B. Grébert, Birkhoff normal form for PDEs with tame modulus, (2004) Zbl1110.37057
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  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), 1-61 Zbl0990.35119MR1833089
  5. J.-M. Delort, J. Szeftel, Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds Zbl1108.58023
  6. J.-M. Delort, J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not. 37 (2004), 1897-1966 Zbl1079.35070MR2056326
  7. M. S. P. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press (1973) Zbl0287.34016
  8. S. Klainerman, The null condition and global existence to nonlinear wave equations, Lectures in Applied Mathematics 23 (1986), 293-326 Zbl0599.35105MR837683
  9. K. Moriyama, Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one-space dimension, Diff. Int. Equations 10 (1997), 499-520 Zbl0891.35096MR1744859
  10. J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685-696 Zbl0597.35101MR803256
  11. H. Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, (2004) Zbl1107.35087MR2228565

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