Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.
Soit, en dimension 3, un système d’équations de Klein-Gordon dont les vitesses sont différentes, avec des termes non-linéaires quadratiques. On montre, pour des données suffisamment petites, regulières et localisées, qu’une solution globale existe et qu’elle disperse. La preuve repose sur la méthode des résonances en espace-temps. La structure des résonances du système se trouve être d’un type qui n’avait pas été étudié jusqu’ici, mais qui est générique dans un certain sens.
Keywords: Klein-Gordon, global existence, resonances
Mots-clés : Klein-Gordon, existence globale, résonances
Germain, Pierre 1
@article{AIF_2011__61_6_2463_0,
author = {Germain, Pierre},
title = {Global existence for coupled {Klein-Gordon} equations with different speeds},
journal = {Annales de l'Institut Fourier},
pages = {2463--2506},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {61},
number = {6},
year = {2011},
doi = {10.5802/aif.2680},
mrnumber = {2976318},
zbl = {1255.35162},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2680/}
}
TY - JOUR AU - Germain, Pierre TI - Global existence for coupled Klein-Gordon equations with different speeds JO - Annales de l'Institut Fourier PY - 2011 SP - 2463 EP - 2506 VL - 61 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2680/ DO - 10.5802/aif.2680 LA - en ID - AIF_2011__61_6_2463_0 ER -
%0 Journal Article %A Germain, Pierre %T Global existence for coupled Klein-Gordon equations with different speeds %J Annales de l'Institut Fourier %D 2011 %P 2463-2506 %V 61 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2680/ %R 10.5802/aif.2680 %G en %F AIF_2011__61_6_2463_0
Germain, Pierre. Global existence for coupled Klein-Gordon equations with different speeds. Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2463-2506. doi : 10.5802/aif.2680. https://aif.centre-mersenne.org/articles/10.5802/aif.2680/
[1] Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., Volume 39 (1986) no. 2, pp. 267-282 | DOI | MR | Zbl
[2] Au delà des opérateurs pseudo-différentiels, Astérisque, 57, Société Mathématique de France, Paris, 1978 | MR | Zbl
[3] Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data, Comm. Partial Differential Equations, Volume 25 (2000) no. 11-12, p. 2119-1269 | DOI | MR | Zbl
[4] Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., Volume 211 (2004) no. 2, pp. 288-323 | DOI | MR | Zbl
[5] Global solutions for 2D quadratic Schrödinger equations. (preprint)
[6] Global solutions for the gravity water waves equation in dimension 3 (preprint) | Zbl
[7] Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN (2009) no. 3, pp. 414-432 | MR | Zbl
[8] Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., Volume 43 (1985) no. 4, pp. 399-442 | Numdam | MR | Zbl
[9] Nonlinear scattering for a system of nonlinear Klein-Gordon equations, J. Math. Phys., Volume 49 (2008) no. 10 | DOI | MR | Zbl
[10] Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin), 26, Springer-Verlag, Berlin, 1997 | MR | Zbl
[11] Scattering threshold for the focusing nonlinear Klein-Gordon equation (arXiv:1001.1474) | Zbl
[12] Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., Volume 34 (1981) no. 1, pp. 29-51 | DOI | MR | Zbl
[13] Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, Osaka J. Math., Volume 43 (2006) no. 2, pp. 283-326 | MR | Zbl
[14] The null condition and global existence to nonlinear wave equations (Nonlinear systems of partial differential equations in applied mathematics Part 1 (Santa Fe, N.M., 1984), p. 293–326, Lectures in Appl. Math.), Volume 23, Amer. Math. Soc., Providence, RI, 1983 | MR | Zbl
[15] Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., Volume 38 (1985) no. 5, pp. 631-641 | DOI | MR | Zbl
[16] Counterexample to global existence for systems of nonlinear wave equations with different propagation speeds, Funkcial. Ekvac., Volume 46 (2003) no. 3, pp. 471-477 | DOI | MR | Zbl
[17] Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., Volume 38 (1985) no. 5, pp. 685-696 | DOI | MR | Zbl
[18] Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal., Volume 33 (2001) no. 2, pp. 477-488 | DOI | MR | Zbl
[19] Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac., Volume 46 (2003) no. 1, pp. 41-62 | DOI | MR | Zbl
[20] Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, Volume 52 (2000) no. 3, pp. 609-632 | DOI | MR | Zbl
Cited by Sources:
