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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 and 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. 

[2]

A. Bachelot, Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon, Ann. Inst. Henri Poincaré, 48 (1988), 387-422. 

[3]

D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. 

[4]

V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698. 

[5]

V. Georgiev, Decay estimates for the Klein-Gordon equations, Comm. Partial Differential Equations, 17 (1992), 1111-1139. 

[6]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématique & Applications 26, Springer-Verlag, Berlin, 1997.

[7]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. 

[8]

F. John, Blow-up of solutions for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51. 

[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. 

[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. 

[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. 

[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. 

[13]

S. KatayamaT. 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. 

[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. 

[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. 

[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.

[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. 

[18]

R. Kosecki, The unit condition and global existence for a class of nonlinear Klein-Gordon equations, J. Differential Equations, 100 (1992), 257-268. 

[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. 

[20]

P. G. LeFloch and Y. Ma, The Hyperboloidal Foliation Method, World Scientific Publishing Co. Pte. Ltd., Singapore, 2015.

[21]

H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472. 

[22]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. 

[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. 

[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. 

show all references

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. 

[2]

A. Bachelot, Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon, Ann. Inst. Henri Poincaré, 48 (1988), 387-422. 

[3]

D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. 

[4]

V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698. 

[5]

V. Georgiev, Decay estimates for the Klein-Gordon equations, Comm. Partial Differential Equations, 17 (1992), 1111-1139. 

[6]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématique & Applications 26, Springer-Verlag, Berlin, 1997.

[7]

F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. 

[8]

F. John, Blow-up of solutions for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51. 

[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. 

[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. 

[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. 

[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. 

[13]

S. KatayamaT. 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. 

[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. 

[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. 

[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.

[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. 

[18]

R. Kosecki, The unit condition and global existence for a class of nonlinear Klein-Gordon equations, J. Differential Equations, 100 (1992), 257-268. 

[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. 

[20]

P. G. LeFloch and Y. Ma, The Hyperboloidal Foliation Method, World Scientific Publishing Co. Pte. Ltd., Singapore, 2015.

[21]

H. Lindblad, On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472. 

[22]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. 

[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. 

[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. 

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