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July  2018, 17(4): 1479-1497. doi: 10.3934/cpaa.2018071

## Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data

 Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, JAPAN

Received  June 2017 Revised  November 2017 Published  April 2018

Fund Project: The author is supported by JSPS, Grant-in-Aid for Scientific Research (C) (JSPS KAKENHI Grant Number JP26400168).

We consider the Cauchy problem for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions. The author previously proved the small data global existence for rapidly decreasing data under a certain condition on nonlinearity. In this paper, we show that we can weaken the condition, provided that the initial data are compactly supported.

Citation: Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071
##### References:
 [1] S. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487.   Google Scholar [2] A. Bachelot, Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon, Ann. Inst. Henri Poincaré, 48 (1988), 387-422.   Google Scholar [3] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282.   Google Scholar [4] V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698.   Google Scholar [5] V. Georgiev, Decay estimates for the Klein-Gordon equations, Comm. Partial Differential Equations, 17 (1992), 1111-1139.   Google Scholar [6] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématique & Applications 26, Springer-Verlag, Berlin, 1997. Google Scholar [7] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.   Google Scholar [8] F. John, Blow-up of solutions for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.   Google Scholar [9] S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213.   Google Scholar [10] S. Katayama, Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions, Math. Z., 270 (2012), 487-513.   Google Scholar [11] S. Katayama, Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions, J. Differential Equations, 255 (2013), 120-150.   Google Scholar [12] S. Katayama and H. Kubo, An alternative proof of global existence for nonlinear wave equations in an exterior domain, J. Math. Soc. Japan, 60 (2008), 1135-1170.   Google Scholar [13] S. Katayama, T. Ozawa and H. Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, Comm. Pure Appl. Math., 65 (2012), 1285-1302.   Google Scholar [14] S. Katayama and K. Yokoyama, Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, Osaka J. Math., 43 (2006), 283-326.   Google Scholar [15] S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641.   Google Scholar [16] S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math., 23 (1986), AMS, Providence, 293-326. Google Scholar [17] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ${\mathbf R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117.   Google Scholar [18] R. Kosecki, The unit condition and global existence for a class of nonlinear Klein-Gordon equations, J. Differential Equations, 100 (1992), 257-268.   Google Scholar [19] K. Kubota and K. Yokoyama, Global existence of classical solutions to systems of nonlinear wave equations with different speeds of propagation, Japan. J. Math. (N.S.), 27 (2001), 113-202.   Google Scholar [20] P. G. LeFloch and Y. Ma, The Hyperboloidal Foliation Method, World Scientific Publishing Co. Pte. Ltd., Singapore, 2015. Google Scholar [21] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472.   Google Scholar [22] J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.   Google Scholar [23] H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325.   Google Scholar [24] Y. Tsutsumi, Global solutions for the Dirac-Proca equations with small initial data in $3+1$ space time dimensions, J. Math. Anal. Appl., 278 (2003), 485-499.   Google Scholar

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##### References:
 [1] S. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487.   Google Scholar [2] A. Bachelot, Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon, Ann. Inst. Henri Poincaré, 48 (1988), 387-422.   Google Scholar [3] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282.   Google Scholar [4] V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698.   Google Scholar [5] V. Georgiev, Decay estimates for the Klein-Gordon equations, Comm. Partial Differential Equations, 17 (1992), 1111-1139.   Google Scholar [6] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématique & Applications 26, Springer-Verlag, Berlin, 1997. Google Scholar [7] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.   Google Scholar [8] F. John, Blow-up of solutions for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.   Google Scholar [9] S. Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213.   Google Scholar [10] S. Katayama, Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions, Math. Z., 270 (2012), 487-513.   Google Scholar [11] S. Katayama, Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions, J. Differential Equations, 255 (2013), 120-150.   Google Scholar [12] S. Katayama and H. Kubo, An alternative proof of global existence for nonlinear wave equations in an exterior domain, J. Math. Soc. Japan, 60 (2008), 1135-1170.   Google Scholar [13] S. Katayama, T. Ozawa and H. Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, Comm. Pure Appl. Math., 65 (2012), 1285-1302.   Google Scholar [14] S. Katayama and K. Yokoyama, Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, Osaka J. Math., 43 (2006), 283-326.   Google Scholar [15] S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641.   Google Scholar [16] S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math., 23 (1986), AMS, Providence, 293-326. Google Scholar [17] S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ${\mathbf R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117.   Google Scholar [18] R. Kosecki, The unit condition and global existence for a class of nonlinear Klein-Gordon equations, J. Differential Equations, 100 (1992), 257-268.   Google Scholar [19] K. Kubota and K. Yokoyama, Global existence of classical solutions to systems of nonlinear wave equations with different speeds of propagation, Japan. J. Math. (N.S.), 27 (2001), 113-202.   Google Scholar [20] P. G. LeFloch and Y. Ma, The Hyperboloidal Foliation Method, World Scientific Publishing Co. Pte. Ltd., Singapore, 2015. Google Scholar [21] H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472.   Google Scholar [22] J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.   Google Scholar [23] H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differential Equations, 192 (2003), 308-325.   Google Scholar [24] Y. Tsutsumi, Global solutions for the Dirac-Proca equations with small initial data in $3+1$ space time dimensions, J. Math. Anal. Appl., 278 (2003), 485-499.   Google Scholar
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