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

Milena Dimova; Natalia Kolkovska; Nikolai Kutev; Milena Dimova; Natalia Kolkovska; Nikolai Kutev

doi:10.3934/era.2020035

Electronic Research Archive

2020, Volume 28, Issue 2: 671-689. doi: 10.3934/era.2020035

Previous Article Next Article

Special Issues

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

Received: 01 December 2019 Revised: 01 March 2020
Primary: 35L70, 35B44; Secondary: 35B40

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.
- Klein-Gordon equation,
- finite time blow up,
- global existence,
- necessary and sufficient conditions,
- critical initial energy
Citation: Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy[J]. Electronic Research Archive, 2020, 28(2): 671-689. doi: 10.3934/era.2020035

Related Papers:

Abstract

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.

References

[1]	Uniform estimates for solutions of nonlinear Klein-Gordon equations. J. Funct. Anal. (1985) 60: 36-55.
[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.
[3]	Revised concavity method and application to Klein-Gordon equation. FILOMAT (2016) 30: 831-839.
[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.
[5]	Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire (2006) 23: 185-207.
[6]	The global Cauchy problem for the non linear Klein-Gordon equation. Math Z. (1985) 189: 487-505.
[7]	The global Cauchy problem for the non linear Klein-Gordon equation. Ⅱ. Ann. Inst. H. Poincaré Anal. Non Linéaire (1989) 6: 15-35.
[8]	Remarks on the qualitative behavior of the undamped Klein-Gordon equation. Math. Methods Appl. Sci. (2018) 41: 103-111.
[9]	Blow up and asymptotic behavior in a nondissipative nonlinear wave equation. Appl. Anal. (2014) 93: 1963-1978.
[10]	The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. J. Soviet Math. (1978) 10: 53-70.
[11]	Blowup of a positive-energy solution of model wave equations in nonlinear dynamics. Theoretical and Mathematical Physics (2012) 171: 421-434.
[12]	Nonexistence of global solutions to new ordinary differential inequality and applications to nonlinear dispersive equations. Math. Methods Appl. Sci. (2016) 39: 2287-2297.
[13]	Sign-preserving functionals and blow-up to Klein-Gordon equation with arbitrary high energy. Appl. Anal. (2016) 95: 860-873.
[14]	T. D. Lee, Particle Physics and Introduction to Field Theory, Contemporary Concepts in Physics, 1. Harwood Academic Publishers, Chur, 1981.
[15]	Instability and nonexistence of global solutions to nonlinear wave equations of the form $Putt=-Au+F(u)$. Trans. Amer. Math. Soc. (1974) 192: 1-21.
[16]	Existence and nonexistence of global solutions for the equation of dislocation of crystals. J. Differntial Equations (1998) 146: 5-21.
[17]	Potential well method for Cauchy problem of generalized double dispersion equations. J. Math. Anal. Appl. (2008) 338: 1169-1187.
[18]	Sharp threshold of global existence and blow-up of the combined nonlinear Klein-Gordon equation. J. Math. Anal. Appl. (2019) 474: 814-832.
[19]	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 (2019) 141: 102-123.
[20]	Saddle points and instability of nonlinear hyperbolic equations. Israel Journal of Mathematics (1975) 22: 273-303.
[21]	Stable standing waves of nonlinear Klein-Gordon equations. Commun. Math. Phys. (1983) 91: 313-327.
[22]	Further global nonexistence theorems for abstract nonlinear wave equations. Proc. Amer. Math. Soc. (1975) 48: 381-390.
[23]	A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy. Proc. Amer. Math. Soc. (2008) 136: 3477-3482.
[24]	Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term. Math. Methods Appl. Sci. (2010) 33: 831-844.
[25]	Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities. Math. Meth. Appl. Sci. (2011) 34: 2318-2328.
[26]	Initial boundary value problem for semilinear hyperbolic equations with critical initial data. Quarterly of Applied Mathematics (2010) 68: 459-468.
[27]	Global solutions and finite time blow up for damped Klein-Gordon equation. Acta Math. Scienta (2013) 33): 643-652.
[28]	Nonlinear wave equation and reaction-diffusion equations with several nonlinear source terms of different signs at high energy level. ANZIAM J. (2013) 54: 153-170.
[29]	Finite time blowup for nonlinear Klein-Gordon equations with arbitrarily positive initial energy. Appl. Math. Letters (2018) 77: 21-26.
[30]	Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations. Nonlinear Anal. (2002) 48: 191-207.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)