Global existence for coupled Klein-Gordon equations with different speeds  [ Solutions globales pour des équations de Klein-Gordon avec des vitesses différentes couplées ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 6, p. 2463-2506
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.
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.
DOI : https://doi.org/10.5802/aif.2680
Classification:  35L70,  47H60
Mots clés: Klein-Gordon, existence globale, résonances
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     pages = {2463-2506},
     doi = {10.5802/aif.2680},
     mrnumber = {2976318},
     zbl = {1255.35162},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_6_2463_0}
}
Germain, Pierre. Global existence for coupled Klein-Gordon equations with different speeds. Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2463-2506. doi : 10.5802/aif.2680. http://www.numdam.org/item/AIF_2011__61_6_2463_0/

[1] Christodoulou, D. Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., Tome 39 (1986) no. 2, pp. 267-282 | Article | MR 820070 | Zbl 0612.35090

[2] Coifman, R.; Meyer, Y. Au delà des opérateurs pseudo-différentiels, Société Mathématique de France, Paris, Astérisque, Tome 57 (1978) | MR 518170 | Zbl 0483.35082

[3] Delort, J.-M.; Fang, D. Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data, Comm. Partial Differential Equations, Tome 25 (2000) no. 11-12, p. 2119-1269 | Article | MR 1789923 | Zbl 0979.35101

[4] Delort, J.-M.; Fang, D.; Xue, R. Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., Tome 211 (2004) no. 2, pp. 288-323 | Article | MR 2056833 | Zbl 1061.35089

[5] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for 2D quadratic Schrödinger equations. (preprint)

[6] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3 (preprint) | Zbl 1177.35168

[7] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN (2009) no. 3, pp. 414-432 | MR 2482120 | Zbl 1156.35087

[8] Ginibre, J.; Velo, G. Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., Tome 43 (1985) no. 4, pp. 399-442 | Numdam | MR 824083 | Zbl 0595.35089

[9] Hayashi, N.; Naumkin, P.; Wibowo, R. Nonlinear scattering for a system of nonlinear Klein-Gordon equations, J. Math. Phys., Tome 49 (2008) no. 10 | Article | MR 2464621 | Zbl 1152.81467

[10] Hörmander, L. Lectures on nonlinear hyperbolic differential equations, Springer-Verlag, Berlin, Mathématiques & Applications (Berlin), Tome 26 (1997) | MR 1466700 | Zbl 0881.35001

[11] Ibrahim, S.; Masmoudi, N.; Nakanishi, K. Scattering threshold for the focusing nonlinear Klein-Gordon equation (arXiv:1001.1474) | Zbl 1270.35132

[12] John, F. Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., Tome 34 (1981) no. 1, pp. 29-51 | Article | MR 600571 | Zbl 0453.35060

[13] Katayama, S.; Yokoyama, K. Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, Osaka J. Math., Tome 43 (2006) no. 2, pp. 283-326 | MR 2262337 | Zbl 1195.35225

[14] Klainerman, S. The null condition and global existence to nonlinear wave equations, Amer. Math. Soc., Providence, RI (Nonlinear systems of partial differential equations in applied mathematics Part 1 (Santa Fe, N.M., 1984), p. 293–326, Lectures in Appl. Math.) Tome 23 (1983) | MR 837683 | Zbl 0599.35105

[15] Klainerman, S. Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., Tome 38 (1985) no. 5, pp. 631-641 | Article | MR 803252 | Zbl 0597.35100

[16] Ohta, M. Counterexample to global existence for systems of nonlinear wave equations with different propagation speeds, Funkcial. Ekvac., Tome 46 (2003) no. 3, pp. 471-477 | Article | MR 2035450 | Zbl 1330.35253 | Zbl pre02112812

[17] Shatah, J. Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., Tome 38 (1985) no. 5, pp. 685-696 | Article | MR 803256 | Zbl 0597.35101

[18] Sideris, T.; Tu, S.-Y. Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal., Tome 33 (2001) no. 2, pp. 477-488 | Article | MR 1857981 | Zbl 1002.35091

[19] Tsutsumi, Y. Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac., Tome 46 (2003) no. 1, pp. 41-62 | Article | MR 1996293 | Zbl 1151.58303

[20] Yokoyama, K. Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, Tome 52 (2000) no. 3, pp. 609-632 | Article | MR 1760608 | Zbl 0968.35081