June  2020, 28(2): 671-689. doi: 10.3934/era.2020035

Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy

1. 

University of National and World Economy, 1700 Sofia, Bulgaria

2. 

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl.8, 1113 Sofia, Bulgaria

* Corresponding author: Milena Dimova

Received  December 2019 Revised  March 2020 Published  April 2020

Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

Citation: 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
References:
[1]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.  Google Scholar

[2]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[3]

M. DimovaK. Kolkovska and N. Kutev, Revised concavity method and application to Klein-Gordon equation, FILOMAT, 30 (2016), 831-839.  doi: 10.2298/FIL1603831D.  Google Scholar

[4]

M. Dimova, N. Kolkovska and N. Kutev, Blow up of solutions to ordinary differential equations arising in nonlinear dispersive problems, Electron. J. of Differential Equations, 2018 (2018), 16 pp.  Google Scholar

[5]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

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J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation, Math Z., 189 (1985), 487-505.  doi: 10.1007/BF01168155.  Google Scholar

[7]

J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15-35.  doi: 10.1016/S0294-1449(16)30329-8.  Google Scholar

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J. A. Esquivel-Avila, Remarks on the qualitative behavior of the undamped Klein-Gordon equation, Math. Methods Appl. Sci., 41 (2018), 103-111.  doi: 10.1002/mma.4598.  Google Scholar

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J. A. Esquivel-Avila, Blow up and asymptotic behavior in a nondissipative nonlinear wave equation, Appl. Anal., 93 (2014), 1963-1978.  doi: 10.1080/00036811.2013.859250.  Google Scholar

[10]

V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, J. Soviet Math., 10 (1978), 53-70.   Google Scholar

[11]

M. O. Korpusov, Blowup of a positive-energy solution of model wave equations in nonlinear dynamics, Theoretical and Mathematical Physics, 171 (2012), 421-434.  doi: 10.1007/s11232-012-0041-6.  Google Scholar

[12]

N. KutevN. Kolkovska and M. Dimova, Nonexistence of global solutions to new ordinary differential inequality and applications to nonlinear dispersive equations, Math. Methods Appl. Sci., 39 (2016), 2287-2297.  doi: 10.1002/mma.3639.  Google Scholar

[13]

N. KutevN. Kolkovska and M. Dimova, Sign-preserving functionals and blow-up to Klein-Gordon equation with arbitrary high energy, Appl. Anal., 95 (2016), 860-873.  doi: 10.1080/00036811.2015.1038994.  Google Scholar

[14]

T. D. Lee, Particle Physics and Introduction to Field Theory, Contemporary Concepts in Physics, 1. Harwood Academic Publishers, Chur, 1981.  Google Scholar

[15]

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

[16]

K. T. Li and Q. D. Zhang, Existence and nonexistence of global solutions for the equation of dislocation of crystals, J. Differntial Equations, 146 (1998), 5-21.  doi: 10.1006/jdeq.1998.3409.  Google Scholar

[17]

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.  Google Scholar

[18]

J. Lu and Q. Y. Miao, Sharp threshold of global existence and blow-up of the combined nonlinear Klein-Gordon equation, J. Math. Anal. Appl., 474 (2019), 814-832.  doi: 10.1016/j.jmaa.2019.01.058.  Google Scholar

[19]

Y. B. LuoY. B. YangM. S. AhmedT. YuM. Y. ZhangL. G. Wang and H. C. Xu, Global existence and blow up of the solution for nonlinear Klein-Gordon equation with general power-type nonlinearities at three initial energy levels, Applied Numerical Mathematics, 141 (2019), 102-123.  doi: 10.1016/j.apnum.2018.05.018.  Google Scholar

[20]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[21]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys., 91 (1983), 313-327.  doi: 10.1007/BF01208779.  Google Scholar

[22]

B. Straughan, Further global nonexistence theorems for abstract nonlinear wave equations, Proc. Amer. Math. Soc., 48 (1975), 381-390.  doi: 10.1090/S0002-9939-1975-0365265-9.  Google Scholar

[23]

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.  Google Scholar

[24]

R.-T. Xu, Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term, Math. Methods Appl. Sci., 33 (2010), 831-844.   Google Scholar

[25]

R. Z. Xu, Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities, Math. Meth. Appl. Sci., 34 (2011), 2318-2328.  doi: 10.1002/mma.1536.  Google Scholar

[26]

R. Z. Xu, Initial boundary value problem for semilinear hyperbolic equations with critical initial data, Quarterly of Applied Mathematics, 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[27]

R. Z. Xu and Y. H. Ding, Global solutions and finite time blow up for damped Klein-Gordon equation, Acta Math. Scienta, 33) (2013), 643-652.  doi: 10.1016/S0252-9602(13)60027-2.  Google Scholar

[28]

R. Z. XuY. B. YangS. H. ChenJ. SuJ. H. Shen and S. B. Huang, Nonlinear wave equation and reaction-diffusion equations with several nonlinear source terms of different signs at high energy level, ANZIAM J., 54 (2013), 153-170.  doi: 10.1017/S1446181113000175.  Google Scholar

[29]

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

[30]

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

show all references

References:
[1]

T. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60 (1985), 36-55.  doi: 10.1016/0022-1236(85)90057-6.  Google Scholar

[2]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[3]

M. DimovaK. Kolkovska and N. Kutev, Revised concavity method and application to Klein-Gordon equation, FILOMAT, 30 (2016), 831-839.  doi: 10.2298/FIL1603831D.  Google Scholar

[4]

M. Dimova, N. Kolkovska and N. Kutev, Blow up of solutions to ordinary differential equations arising in nonlinear dispersive problems, Electron. J. of Differential Equations, 2018 (2018), 16 pp.  Google Scholar

[5]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[6]

J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation, Math Z., 189 (1985), 487-505.  doi: 10.1007/BF01168155.  Google Scholar

[7]

J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15-35.  doi: 10.1016/S0294-1449(16)30329-8.  Google Scholar

[8]

J. A. Esquivel-Avila, Remarks on the qualitative behavior of the undamped Klein-Gordon equation, Math. Methods Appl. Sci., 41 (2018), 103-111.  doi: 10.1002/mma.4598.  Google Scholar

[9]

J. A. Esquivel-Avila, Blow up and asymptotic behavior in a nondissipative nonlinear wave equation, Appl. Anal., 93 (2014), 1963-1978.  doi: 10.1080/00036811.2013.859250.  Google Scholar

[10]

V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, J. Soviet Math., 10 (1978), 53-70.   Google Scholar

[11]

M. O. Korpusov, Blowup of a positive-energy solution of model wave equations in nonlinear dynamics, Theoretical and Mathematical Physics, 171 (2012), 421-434.  doi: 10.1007/s11232-012-0041-6.  Google Scholar

[12]

N. KutevN. Kolkovska and M. Dimova, Nonexistence of global solutions to new ordinary differential inequality and applications to nonlinear dispersive equations, Math. Methods Appl. Sci., 39 (2016), 2287-2297.  doi: 10.1002/mma.3639.  Google Scholar

[13]

N. KutevN. Kolkovska and M. Dimova, Sign-preserving functionals and blow-up to Klein-Gordon equation with arbitrary high energy, Appl. Anal., 95 (2016), 860-873.  doi: 10.1080/00036811.2015.1038994.  Google Scholar

[14]

T. D. Lee, Particle Physics and Introduction to Field Theory, Contemporary Concepts in Physics, 1. Harwood Academic Publishers, Chur, 1981.  Google Scholar

[15]

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

[16]

K. T. Li and Q. D. Zhang, Existence and nonexistence of global solutions for the equation of dislocation of crystals, J. Differntial Equations, 146 (1998), 5-21.  doi: 10.1006/jdeq.1998.3409.  Google Scholar

[17]

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.  Google Scholar

[18]

J. Lu and Q. Y. Miao, Sharp threshold of global existence and blow-up of the combined nonlinear Klein-Gordon equation, J. Math. Anal. Appl., 474 (2019), 814-832.  doi: 10.1016/j.jmaa.2019.01.058.  Google Scholar

[19]

Y. B. LuoY. B. YangM. S. AhmedT. YuM. Y. ZhangL. G. Wang and H. C. Xu, Global existence and blow up of the solution for nonlinear Klein-Gordon equation with general power-type nonlinearities at three initial energy levels, Applied Numerical Mathematics, 141 (2019), 102-123.  doi: 10.1016/j.apnum.2018.05.018.  Google Scholar

[20]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[21]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys., 91 (1983), 313-327.  doi: 10.1007/BF01208779.  Google Scholar

[22]

B. Straughan, Further global nonexistence theorems for abstract nonlinear wave equations, Proc. Amer. Math. Soc., 48 (1975), 381-390.  doi: 10.1090/S0002-9939-1975-0365265-9.  Google Scholar

[23]

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.  Google Scholar

[24]

R.-T. Xu, Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term, Math. Methods Appl. Sci., 33 (2010), 831-844.   Google Scholar

[25]

R. Z. Xu, Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities, Math. Meth. Appl. Sci., 34 (2011), 2318-2328.  doi: 10.1002/mma.1536.  Google Scholar

[26]

R. Z. Xu, Initial boundary value problem for semilinear hyperbolic equations with critical initial data, Quarterly of Applied Mathematics, 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[27]

R. Z. Xu and Y. H. Ding, Global solutions and finite time blow up for damped Klein-Gordon equation, Acta Math. Scienta, 33) (2013), 643-652.  doi: 10.1016/S0252-9602(13)60027-2.  Google Scholar

[28]

R. Z. XuY. B. YangS. H. ChenJ. SuJ. H. Shen and S. B. Huang, Nonlinear wave equation and reaction-diffusion equations with several nonlinear source terms of different signs at high energy level, ANZIAM J., 54 (2013), 153-170.  doi: 10.1017/S1446181113000175.  Google Scholar

[29]

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

[30]

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

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