[1] Alinhac S. Blowup of small data solutions for a quasilinear wave equation in two space dimensions. Ann Math, 1999, 149(1): 97-127 [2] Alinhac S. Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions II. Acta Math, 1999, 182(1): 1-23 [3] Alinhac S. The null condition for quasilinear wave equations in two-space dimension I. Invent Math, 2001, 145(3): 597-618 [4] Alinhac S. The null condition for quasilinear wave equations in two space dimensions II. Amer J Math, 2001, 123(6): 1071-1101 [5] Alinhac S.Hyperbolic Partial Differential Equations. New York: Springer, 2009 [6] Bourgain J. Construction of approximative and almost periodic solutions of perturbed linear Schr$\mathrm{\ddot{o}}$dinger and wave equations. Geom Funct Anal, 1996, 6(2): 201-230 [7] Bambusi D. Birkhoff normal form for some nonlinear PDEs. Comm Math Phys, 2003, 234(2): 253-285 [8] Bambusi D, Grebert B. Birkhoff normal form for partial differential equations with tame modulus. Duke Math J, 2006, 135(3): 507-567 [9] Christodoulou D. Global solutions of nonlinear hyperbolic equations for small initial data. Commun Pure Appl Math, 1986, 39(2): 267-282 [10] Cai Y, Lei Z, Masmoudi N. Global well-posedness for 2D nonlinear wave equations without compact support. J Math Pures Appl, 2018, 114: 211-234 [11] Delort J M. Temps d'existence pour l'$\acute{\mathrm{e}}$quation de Klein-Gordon semi-lin$\acute{\mathrm{e}}$aire $\acute{\mathrm{a}}$ donn$\acute{\mathrm{e}}$es petites p$\acute{\mathrm{e}}$riodiques. (French) [Time of existence for the semilinear Klein-Gordon equation with periodic small data]. Amer J Math, 1998, 120(3): 663-689 [12] Delort J M. Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire á données petites en dimension 1. (French) [Global existence and asymptotic behavior for the quasilinear Klein-Gordon equation with small data in dimension 1]. Ann Sci École Norm Sup, (4), 2001, 34(1): 1-61 [13] Delort J M, Szeftel J.Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres. Int Math Res Not, 2004, 2004(37): 1897-1966 [14] Ettinger B. Well-posedness of the equation for the three-form field in eleven-dimensional supergravity. Trans Amer Math Soc, 2015, 367(2): 887-910 [15] Godin P. Lifespan of solutions of semilinear wave equations in two space dimensions. Comm Partial Differential Equations, 1993, 18(5/6): 895-916 [16] Georgiev V. Decay estimates for the Klein-Gordon equation. Comm Partial Differential Equations, 1992, 17(7/8): 1111-1139 [17] Georgiev V. Global solution of the system of wave and Klein-Gordon equations. Math Z, 1990, 203(4): 683-698 [18] Hörmander L.Lectures on Nonlinear Hyperbolic Equations. Mathematiques & Applications. Berlin: Springer-Verlag, 1997 [19] Huneau C, Stingo A. Global well-posedness for a system of quasilinear wave equations on a product space. arXiv:2110.13982vl [20] Hari L, Visciglia N. Small data scattering for energy-subcritical and critical Nonlinear Klein Gordon equations on product spaces. arXiv:1603.06762v1 [21] Hari L, Visciglia N. Small data scattering for energy critical NLKG on product spaces~$\mathbb{R}^{d}\times \mathcal{M}^{2}$. Commun Contemp Math, 2018, 20(2): 1750036 [22] Hou F, Yin H C. Global small data smooth solutions of 2-D null-form wave equations with non-compactly supported initial data. J Differential Equations, 2020, 268(2): 490-512 [23] John F. Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math, 1979, 28(1/3): 235-268 [24] John F. Blow-up for quasilinear wave equations in three space dimensions. Comm Pure Appl Math, 1981, 34(1): 29-51 [25] Klein O. Quantum theory and five-dimensional theory of relativity. Z Phys A, 1926, 37: 895-906 [26] Klainerman S, Ponce G. Global small amplitude solutions to nonlinear evolution equations. Comm Pure Appl Math, 1983, 36(1): 133-141 [27] Klainerman S. Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. Comm Pure Appl Math, 1985, 38(5): 631-641 [28] Klainerman S. 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). Lectures in Appl Math Amer Math Soc, 1986, 23: 293-326 [29] Kaluza T. Zum Unit$\ddot{\mathrm{a}}$tsproblem der Physik. Int J Mod Phys D, 2018, 27(14): 1870001 [30] Luli G K, Yang S W, Yu P. On one-dimension semi-linear wave equations with null conditions. Adv Math, 2018, 329: 174-188 [31] Li J, Tao F, Yin H C. Almost global smooth solutions of 3D quasilinear Klein-Gordon equations on the product space $\mathbb{R}^{2}\times \mathbb{T}$. arXiv:2204.08130v1 [32] Lesky P H, Racke R. Nonlinear wave equations in infinite waveguides. Comm Partial Differential Equations, 2003, 28(7/8): 1265-1301 [33] Liu Yingbo, Ingo W. Small data solutions of 2D quasilinear wave equations under null conditions. Acta Math Sci, 2018, 38B(1): 125-150 [34] Lai N A, Schiavone N M. Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture. Math Z, 2022, 301(4): 3369-3393 [35] Metcalfe J, Sogge C D, Stewart A. Nonlinear hyperbolic equations in infinite homogeneous waveguides. Comm Partial Differential Equations, 2005, 30(4/6): 643-661 [36] Ozawa T, Tsutaya K, Tsutsumi Y. Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions. Math Z, 1996, 222(3): 341-362 [37] Shatah J. Global existence of small solutions to nonlinear evolution equations. J Differ Equ, 1982, 46(3): 409-425 [38] Shatah J. Normal forms and quadratic nonlinear Klein-Gordon equations. Comm Pure Appl Math, 1985, 38(5): 685-696 [39] Shatah J, Struwe M. Geometric Wave Equations.Courant Lecture Notes in Mathematics, 2. Providence, RI: American Mathematical Society, 1998 [40] Simon J C H, Taflin E. The Cauchy problem for nonlinear Klein-Gordon equations. Comm Math Phys, 1993, 152(3): 433-478 [41] Tao F, Yin H C. Global smooth solutions of the 4-D quasilinear Klein-Gordon equations on the product space $\mathbb{R}^{3}\times \mathbb{T}$. J Differ Equ, 2023, 352: 67-121 [42] X R Y, Fang D Y. Global existence of solutions for quadratic quasi-linear Klein-Gordon systems in one space dimension. Acta Math Sci, 2005, 25B(2): 340-358 [43] Zha D B. Global and almost global existence for general quasilinear wave equations in two space dimensions. J Math Pures Appl, 2019, 123: 270-299 [44] Zha D B. On one-dimension quasilinear wave equations with null conditions. Calc Var Partial Differential Equations, 2020, 59(3): Art 94 |