### The Existence and Nonexistence of Global Solutions of the Cauchy Problem for Systems of three semilinear Klein–Gordon equations

#### Abstract

Among the nonlinear hyperbolic equations Klein-Gordon equation has important theoretical and practical meaning. The nonlinear Klein–Gordon equation appears in the study of several problems of mathematical physics. For example, this equation arises in general relativity, nonlinear optics (e.g., the instability phenomena such as self-focusing), plasma physics, fluid mechanics, radiation theory or spin waves [1-3].

#### References

ISegal, I.: Nonlinear partial differential equations in quantum field theory. Proc. Symp. Appl. Math. A.M.S. 17, 210–226 , (1965). I. E., Segal, "The global Cauchy problem for a relativistic scalar field with power interaction," Bull. Soc. Math.

France, 91, No. 2, 129-135 (1963).

V.G. Makhankov, Dynamics of classical solutions in integrable systems, Phys. Rep. 35 (1978) 1_128.

L.A. Medeiros, G. Perla Menzala, On a mixed problem for a class of nonlinear Klein_Gordon equations, Acta Math. Hungar. 52 (1988) 61_69.

Lions JL. Quelques Méthodes De Résolution Des Problèmes Aux limites Non Linéaires. Dunod Gauthier-Villars: Paris, 1969.

T.T. Li, Y. Zhou, Breakdown of solutions to , Discrete Contin. Dyn. Syst. 1 (1995) 503–520.

Q.S. Zhag, A blow-up result for a nonlinear wave equation with damping, C. R. Acad. Sci. Paris, Serie I 333 (2001) 109–114. The critical case.

G.Todorova ,B. Yordanov, Critical exponent for a nonlinear wave equation

with damping, C. R. Acad. Sci. Paris, t. 330, Série I, p. 557–562, 2000

B.R. Ikehata, Y. Miataka, Y. Nakataka, Decay estimates of solutions of dissipative wave equations in Rn with lower power nonlinearities, J. Math. Soc.

Japan 56 (2) (2004) 365–373.

D. H. Sattinger, On global solutions for nonlinear hyperbolic equations

Arch. Ration. Mech. Anal. 30, 148–172 (1968).

] L.E. Payne, D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Math. J. 22 (1975) 273_303.

Y.C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations 192 (2003) 155_169.

R.Z. Xu, Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein_Gordon equation with dissipative term, Math. Methods

Appl. Sci. (2009), in press (doi:10.1002/mma.1196).

J. Zhang, On the standing wave in coupled non-linear Klein_Gordon equations, Math. Methods Appl. Sci. 26 (2003) 11_25

Wenjun Liu, Global Existence, Asymptotic Behavior and Blow-up of Solutions for Coupled Klein–Gordon Equations with Damping Terms, Nonlinear Anal.,73, 244–255 (2010).

A. B. Aliev and A. A. Kazimov, The Existence and Nonexistence of Global Solutions of the Cauchy Problem for Klein–Gordon Systems. Dokl. Math. 87 (1), 39–41 (2013).

Aliev, A.B., Kazimov, A.A.: Nonexistence of Global Solutions of the Cauchy Problem for Systems of Klein–Gordon Equations with Positive Initial Energy, Differential Equations, 51 (12), 1563–1568, (2015).

Alvino, P.L.Lions, G. Trombetti, Comparison results for elliptic and parabolic via Schwarz symmetrization, Annales de l’I.H.P, section C, tome 7, Nř2, (1990), 37-65.

S. Kesavan, Symmetrization and applications, Series in Analysis, Vol. 3, (2006).

G. Polya, G. Szeg¨o, Isoperimetric inequalities in mathematical physics, Ann. Math. Stud. Princeton, University Press, 27 (1952)., 645--651 (1970)W.A. : On weak solutions of semi-linear hyperbolic equations. Anais

W. A. Straus, On weak solutions of semi-linear hyperbolic equations, An. Acad. Brazil Ci. 42 (1970), 645-651.

R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975

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