• Previous Article
    Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions
  • CPAA Home
  • This Issue
  • Next Article
    Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type
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. 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.   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

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.   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. 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.   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

[1]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973

[2]

Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy. Electronic Research Archive, 2020, 28 (2) : 671-689. doi: 10.3934/era.2020035

[3]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407

[4]

Guangyu Xu, Chunlai Mu, Dan Li. Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2491-2512. doi: 10.3934/cpaa.2020109

[5]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903

[6]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679

[7]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[8]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359

[9]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251

[10]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215

[11]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233

[12]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

[13]

Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085

[14]

Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279

[15]

Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043

[16]

Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024

[17]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235

[18]

Hideo Kubo. Global existence for exterior problems of semilinear wave equations with the null condition in $2$D. Evolution Equations & Control Theory, 2013, 2 (2) : 319-335. doi: 10.3934/eect.2013.2.319

[19]

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133

[20]

Marcelo M. Cavalcanti, Leonel G. Delatorre, Daiane C. Soares, Victor Hugo Gonzalez Martinez, Janaina P. Zanchetta. Uniform stabilization of the Klein-Gordon system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5131-5156. doi: 10.3934/cpaa.2020230

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (99)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]