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May  2020, 19(5): 2491-2512. doi: 10.3934/cpaa.2020109

Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

2. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China

* Corresponding author

Received  October 2018 Revised  September 2019 Published  March 2020

This paper considers the Cauchy problem for a nonlinear Klein-Gordon system with damping terms. In the existing works, the solution with low and critical initial energy was studied. We extend the previous results on following three aspects. Firstly, we consider the vacuum isolating phenomenon of solution under initial energy $ E(0)\leq0 $. We find that the corresponding vacuum region is an ball and it expands to whole phase space as $ E(0) $ decays to $ -\infty $. Secondly, we discuss the asymptotic behavior of blow-up solution and prove that the solution grows exponentially. The growth speed is estimated especially. Finally, the solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the solution exists globally and blows up in finite time are given, respectively.

Citation: 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 and Applied Analysis, 2020, 19 (5) : 2491-2512. doi: 10.3934/cpaa.2020109
References:
[1]

A. B. Aliev and A. A. Kazimov, The existence and nonexistence of global solutions of the Cauchy problem for Klein-Gordon systems, Dokl. Math., 90 (2014), 680-682.  doi: 10.1134/s1064562414070084.

[2]

A. B. Aliev and A. A. Kazimov, Nonexistence of global solutions of the Cauchy problem for systems of Klein-Gordon equations with positive initial energy, Differ. Equ., 51 (2015), 1563-1568.  doi: 10.1134/S0012266115120034.

[3]

A. B. Aliev and G. I. Yusifova, Nonexistecne of global solutions of Cauchy problem for systems of semilinear hyperbolic equations with positive initial energy, Electron. J. Differ. Equ., 211 (2017), 1-10. 

[4]

C. O. AlvesM. M. CavalcantiV. N. D. CavalcantiM. Rammaha and D. Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 583-608.  doi: 10.3934/dcdss.2009.2.583.

[5]

M. AlimohammadyC. Cattani and M. K. Kalleji, Invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity, J. Math. Anal. Appl., 455 (2017), 569-591.  doi: 10.1016/j.jmaa.2017.05.057.

[6]

J. DelortD. Fang and R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323.  doi: 10.1016/j.jfa.2004.01.008.

[7]

H. Chen and G. W. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349.  doi: 10.1007/s11868-012-0046-9.

[8]

D. Fang and R. Xue, Global existence of small solutions for cubic quasi-linear Klein-Gordon systems in one space dimension, Acta. Math. Sin. English Ser., 22 (2006), 1085-1102.  doi: 10.1007/s10114-005-0668-4.

[9]

V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698.  doi: 10.1007/BF02570764.

[10]

Y. Q. Guo and M. A. Rammaha, Blow-up of solutions to systems of nonlinear wave equations with supercritical sources, Appl. Anal., 92 (2013), 1101-1115.  doi: 10.1080/00036811.2011.649734.

[11]

F. John, Blow-up for quasilinear wave equations in three space dimensions, Commun. Pure Appl. Math., 34 (2010), 29-51.  doi: 10.1002/cpa.3160340103.

[12]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differ. Equ., 251 (2011), 2549-2567.  doi: 10.1016/j.jde.2011.04.001.

[13]

N. KutevN. Kolkovska and M. Dimova, Global existence of cauchy problem for Boussinesq paradigm equation, Comput. Math. Appl., 65 (2013), 500-511.  doi: 10.1016/j.camwa.2012.05.024.

[14]

C. M. Li and E. S. Wright, Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics, Commun. Pure Appl. Anal., 1 (2017), 77-84.  doi: 10.3934/cpaa.2002.1.77.

[15]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $pu_tt = -au+f(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.

[16]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.

[17]

M. R. Li and L. Y. Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal. Theory Methods Appl., 54 (2003), 1397-1415.  doi: 10.1016/S0362-546X(03)00192-5.

[18]

M. R. Li and L. Y. Tsai, On a system of nonlinear wave equations, Taiwan. J. Math., 7 (2003), 557-573.  doi: 10.11650/twjm/1500407577.

[19]

W. J. Liu, Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms, Nonlinear Anal. Theory Methods Appl., 73 (2010), 244-255.  doi: 10.1016/j.na.2010.03.017.

[20]

X. L. Li and B. Y. Liu, Vacuum isolating, blow up threshold and asymptotic behavior of solutions for a nonlocal parabolic equation, J. Math. Phys., 58 (2017), 101503. doi: 10.1063/1.5004668.

[21]

Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differ. Equ., 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.

[22]

Y. C. Liu and R. Z. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differ. Equ., 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.

[23]

Y. C. Liu and R. Z. Xu, Global existence and blow up of solutions for cauchy problem of generalized boussinesq equation, Physica D, 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.

[24]

Y. C. Liu and R. Z. Xu, Potential well method for cauchy problem of generalized double dispersion equations, J. Math. Anal. Appl., 338 (2008), 1169-1187.  doi: 10.1016/j.jmaa.2007.05.076.

[25]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. Theory Methods Appl., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.

[26]

M. M. Miranda and L. A. Medeiros, On the existence of global solutions of a coupled nonlinear Klein-Gordon equations, Funkc. Ekvacioj Ser. Int., 30 (1987), 147-161. 

[27]

V. G. Makhankov, Dynamics of classical solitons in non-integrable systems, Phys. Rep. Rev. Sec. Phys. Lett., 35 (1978), 1-128.  doi: 10.1016/0370-1573(78)90074-1.

[28]

L. A. Medeiros and G. P. Menzala, On a mixed problem for a class of nonlinear Klein-Gordon equations, Acta Math. Hung., 52 (1988), 61-69.  doi: 10.1007/BF01952481.

[29]

L. A. Medeiros and M. M. Miranda, Weak solutions for a system of nonlinear klein-gordon equations, Ann. Mat. Pura Appl., 146 (1986), 173-183.  doi: 10.1007/BF01762364.

[30]

S. A. Messaoudi and B. Said-Houari, Global nonexistence of positive initial energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277-287.  doi: 10.1016/j.jmaa.2009.10.050.

[31]

M. Reed, Abstract Nonlinear Wave Equations, Springer, Berlin, 1976.

[32]

B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differ. Integral Equ., 23 (2010), 79-92. 

[33]

B. Said-Houari, Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, Z. Angew. Math. Phys., 62 (2011), 115-133.  doi: 10.1007/s00033-010-0082-3.

[34]

B. Said-Houari, Global existence and decay of solutions of a nonlinear system of wave equations, Appl. Anal., 91 (2012), 475-489.  doi: 10.1080/00036811.2010.549475.

[35]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differ. Equ., 192 (2003), 308-325.  doi: 10.1016/S0022-0396(03)00125-6.

[36]

I. E. Segal, Nonlinear partial differential equations in quantum field theory, in Proc. Sympos. Appl. Math., Vol. XVII, Amer. Math. Soc., Providence, R.I. (1965), 210–226.

[37]

I. E. Segal, Non-linear semi-groups, Ann. Math., 78 (1963), 339-364.  doi: 10.2307/1970347.

[38]

S. B. Wang and X. Su, The Cauchy problem for the dissipative Boussinesq equation, Nonlinear Anal. Real World Appl., 45 (2019), 116-141.  doi: 10.1016/j.nonrwa.2018.06.012.

[39]

S. T. Wu, Global existence, blow-up and asymptotic behavior of solutions for a class of coupled nonlinear klein-gordon equations with damping terms, Acta Appl. Math., 119 (2012), 75-95.  doi: 10.1007/s10440-011-9662-2.

[40]

Y. J. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477-3482.  doi: 10.1090/S0002-9939-08-09514-2.

[41]

Y. J. Wang, Nonexistence of global solutions of a class of coupled nonlinear Klein-Gordon equations with nonnegative potentials and arbitrary initial energy, IMA J. Appl. Math., 24 (2009), 392-415.  doi: 10.1093/imamat/hxp004.

[42]

G. Y. Xu, Global existence, finite time blow-up and vacuum isolating phenomena for semilinear parabolic equation with conical degeneration, Taiwan. J. Math., 22 (2018), 1479-1508.  doi: 10.11650/tjm/180302.

[43]

Y. B. Yang and R. Z. Xu, Finite time blowup for nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Appl. Math. Lett., 77 (2018), 21-26.  doi: 10.1016/j.aml.2017.09.014.

[44]

J. Zhang, Stability fo standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/s000330050011.

[45]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal. Theory Methods Appl., 48 (2002), 191-207.  doi: 10.1016/S0362-546X(00)00180-2.

[46]

J. Zhang, On the standing wave in coupled non-linear Klein-Gordon equations, Math. Meth. Appl. Sci., 26 (2003), 11-25.  doi: 10.1002/mma.340.

[47]

R. ZengC. L. Mu and S. M. Zhou, A blow-up result for Kirchhoff-type equations with high energy, Math. Meth. Appl. Sci., 34 (2011), 479-486.  doi: 10.1002/mma.1374.

show all references

References:
[1]

A. B. Aliev and A. A. Kazimov, The existence and nonexistence of global solutions of the Cauchy problem for Klein-Gordon systems, Dokl. Math., 90 (2014), 680-682.  doi: 10.1134/s1064562414070084.

[2]

A. B. Aliev and A. A. Kazimov, Nonexistence of global solutions of the Cauchy problem for systems of Klein-Gordon equations with positive initial energy, Differ. Equ., 51 (2015), 1563-1568.  doi: 10.1134/S0012266115120034.

[3]

A. B. Aliev and G. I. Yusifova, Nonexistecne of global solutions of Cauchy problem for systems of semilinear hyperbolic equations with positive initial energy, Electron. J. Differ. Equ., 211 (2017), 1-10. 

[4]

C. O. AlvesM. M. CavalcantiV. N. D. CavalcantiM. Rammaha and D. Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 583-608.  doi: 10.3934/dcdss.2009.2.583.

[5]

M. AlimohammadyC. Cattani and M. K. Kalleji, Invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity, J. Math. Anal. Appl., 455 (2017), 569-591.  doi: 10.1016/j.jmaa.2017.05.057.

[6]

J. DelortD. Fang and R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323.  doi: 10.1016/j.jfa.2004.01.008.

[7]

H. Chen and G. W. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349.  doi: 10.1007/s11868-012-0046-9.

[8]

D. Fang and R. Xue, Global existence of small solutions for cubic quasi-linear Klein-Gordon systems in one space dimension, Acta. Math. Sin. English Ser., 22 (2006), 1085-1102.  doi: 10.1007/s10114-005-0668-4.

[9]

V. Georgiev, Global solution of the system of wave and Klein-Gordon equations, Math. Z., 203 (1990), 683-698.  doi: 10.1007/BF02570764.

[10]

Y. Q. Guo and M. A. Rammaha, Blow-up of solutions to systems of nonlinear wave equations with supercritical sources, Appl. Anal., 92 (2013), 1101-1115.  doi: 10.1080/00036811.2011.649734.

[11]

F. John, Blow-up for quasilinear wave equations in three space dimensions, Commun. Pure Appl. Math., 34 (2010), 29-51.  doi: 10.1002/cpa.3160340103.

[12]

Y. Kawahara and H. Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differ. Equ., 251 (2011), 2549-2567.  doi: 10.1016/j.jde.2011.04.001.

[13]

N. KutevN. Kolkovska and M. Dimova, Global existence of cauchy problem for Boussinesq paradigm equation, Comput. Math. Appl., 65 (2013), 500-511.  doi: 10.1016/j.camwa.2012.05.024.

[14]

C. M. Li and E. S. Wright, Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics, Commun. Pure Appl. Anal., 1 (2017), 77-84.  doi: 10.3934/cpaa.2002.1.77.

[15]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $pu_tt = -au+f(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.

[16]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.

[17]

M. R. Li and L. Y. Tsai, Existence and nonexistence of global solutions of some system of semilinear wave equations, Nonlinear Anal. Theory Methods Appl., 54 (2003), 1397-1415.  doi: 10.1016/S0362-546X(03)00192-5.

[18]

M. R. Li and L. Y. Tsai, On a system of nonlinear wave equations, Taiwan. J. Math., 7 (2003), 557-573.  doi: 10.11650/twjm/1500407577.

[19]

W. J. Liu, Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms, Nonlinear Anal. Theory Methods Appl., 73 (2010), 244-255.  doi: 10.1016/j.na.2010.03.017.

[20]

X. L. Li and B. Y. Liu, Vacuum isolating, blow up threshold and asymptotic behavior of solutions for a nonlocal parabolic equation, J. Math. Phys., 58 (2017), 101503. doi: 10.1063/1.5004668.

[21]

Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differ. Equ., 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.

[22]

Y. C. Liu and R. Z. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differ. Equ., 244 (2008), 200-228.  doi: 10.1016/j.jde.2007.10.015.

[23]

Y. C. Liu and R. Z. Xu, Global existence and blow up of solutions for cauchy problem of generalized boussinesq equation, Physica D, 237 (2008), 721-731.  doi: 10.1016/j.physd.2007.09.028.

[24]

Y. C. Liu and R. Z. Xu, Potential well method for cauchy problem of generalized double dispersion equations, J. Math. Anal. Appl., 338 (2008), 1169-1187.  doi: 10.1016/j.jmaa.2007.05.076.

[25]

Y. C. Liu and J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. Theory Methods Appl., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011.

[26]

M. M. Miranda and L. A. Medeiros, On the existence of global solutions of a coupled nonlinear Klein-Gordon equations, Funkc. Ekvacioj Ser. Int., 30 (1987), 147-161. 

[27]

V. G. Makhankov, Dynamics of classical solitons in non-integrable systems, Phys. Rep. Rev. Sec. Phys. Lett., 35 (1978), 1-128.  doi: 10.1016/0370-1573(78)90074-1.

[28]

L. A. Medeiros and G. P. Menzala, On a mixed problem for a class of nonlinear Klein-Gordon equations, Acta Math. Hung., 52 (1988), 61-69.  doi: 10.1007/BF01952481.

[29]

L. A. Medeiros and M. M. Miranda, Weak solutions for a system of nonlinear klein-gordon equations, Ann. Mat. Pura Appl., 146 (1986), 173-183.  doi: 10.1007/BF01762364.

[30]

S. A. Messaoudi and B. Said-Houari, Global nonexistence of positive initial energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277-287.  doi: 10.1016/j.jmaa.2009.10.050.

[31]

M. Reed, Abstract Nonlinear Wave Equations, Springer, Berlin, 1976.

[32]

B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differ. Integral Equ., 23 (2010), 79-92. 

[33]

B. Said-Houari, Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, Z. Angew. Math. Phys., 62 (2011), 115-133.  doi: 10.1007/s00033-010-0082-3.

[34]

B. Said-Houari, Global existence and decay of solutions of a nonlinear system of wave equations, Appl. Anal., 91 (2012), 475-489.  doi: 10.1080/00036811.2010.549475.

[35]

H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. Differ. Equ., 192 (2003), 308-325.  doi: 10.1016/S0022-0396(03)00125-6.

[36]

I. E. Segal, Nonlinear partial differential equations in quantum field theory, in Proc. Sympos. Appl. Math., Vol. XVII, Amer. Math. Soc., Providence, R.I. (1965), 210–226.

[37]

I. E. Segal, Non-linear semi-groups, Ann. Math., 78 (1963), 339-364.  doi: 10.2307/1970347.

[38]

S. B. Wang and X. Su, The Cauchy problem for the dissipative Boussinesq equation, Nonlinear Anal. Real World Appl., 45 (2019), 116-141.  doi: 10.1016/j.nonrwa.2018.06.012.

[39]

S. T. Wu, Global existence, blow-up and asymptotic behavior of solutions for a class of coupled nonlinear klein-gordon equations with damping terms, Acta Appl. Math., 119 (2012), 75-95.  doi: 10.1007/s10440-011-9662-2.

[40]

Y. J. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477-3482.  doi: 10.1090/S0002-9939-08-09514-2.

[41]

Y. J. Wang, Nonexistence of global solutions of a class of coupled nonlinear Klein-Gordon equations with nonnegative potentials and arbitrary initial energy, IMA J. Appl. Math., 24 (2009), 392-415.  doi: 10.1093/imamat/hxp004.

[42]

G. Y. Xu, Global existence, finite time blow-up and vacuum isolating phenomena for semilinear parabolic equation with conical degeneration, Taiwan. J. Math., 22 (2018), 1479-1508.  doi: 10.11650/tjm/180302.

[43]

Y. B. Yang and R. Z. Xu, Finite time blowup for nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Appl. Math. Lett., 77 (2018), 21-26.  doi: 10.1016/j.aml.2017.09.014.

[44]

J. Zhang, Stability fo standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/s000330050011.

[45]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal. Theory Methods Appl., 48 (2002), 191-207.  doi: 10.1016/S0362-546X(00)00180-2.

[46]

J. Zhang, On the standing wave in coupled non-linear Klein-Gordon equations, Math. Meth. Appl. Sci., 26 (2003), 11-25.  doi: 10.1002/mma.340.

[47]

R. ZengC. L. Mu and S. M. Zhou, A blow-up result for Kirchhoff-type equations with high energy, Math. Meth. Appl. Sci., 34 (2011), 479-486.  doi: 10.1002/mma.1374.

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