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On the standing wave in coupled non-linear Klein-Gordon equations. (English) Zbl 1034.35080

The author is concerned with the standing wave in coupled nonlinear Klein-Gordon equations. He uses a constrained minimisation argument to prove the existence of a ground state. Then, he gives a criterion for blowing up in a finite time and global existence in Theorem 2. In Theorem 3, he proves the instability of the standing wave.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L55 Higher-order hyperbolic systems
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[1] Georgiev, Mathematische Zeitschrift 203 pp 683– (1990)
[2] Melrose, Annals of Mathematics 121 pp 187– (1985)
[3] Ozawa, Annales de I’Institut Henri Poincaré 12 pp 459– (1995)
[4] Rauch, Communications in Partial Differential Equations 7 pp 1117– (1982)
[5] Simon, Communications in Mathematical Physics 99 pp 541– (1985)
[6] Tsutsumi, Advances in Mathematical Sciences and Applications 8 pp 691– (1999)
[7] Benjamin, Proceedings of the Royal Society London Series A 328 pp 153– (1972)
[8] Berestycki, Comptes Rendus des Seances de l’ Academie des Sciences 293 pp 489– (1981)
[9] Berestycki, Archive for Rational Mechanics and Analysis 82 pp 313– (1983)
[10] Nonlinear Waves, Solitons and Chaos.Cambridge University Press: Cambridge, 1990. · Zbl 0726.76018
[11] Liu, Journal of Dynamics and Differential Equations pp 537– (1993)
[12] Ohta, Annales de I’Institut Henri Poincaré 62 pp 69– (1995)
[13] Nonlinear Wave Equations. vol. 73.American Mathematical Society: Providence, 1989.
[14] Strauss, Communications in Mathematical Physics 55 pp 149– (1977)
[15] Shatah, Communications in Mathematical Physics 100 pp 173– (1985)
[16] Souganidis, Proceedings of the Royal Society, Edinburgh 114A pp 195– (1990) · Zbl 0713.35108 · doi:10.1017/S0308210500024380
[17] Applied Functional Analysis.Springer: Berlin, 1995. · doi:10.1007/978-1-4612-0815-0
[18] Payne, Israel Journal of Mathematics 22 pp 273– (1975)
[19] Levine, Transactions of the American Mathematical Society 92 pp 1– (1974)
[20] Grozdena, Comptes Rendus des Seances de l’ Academie des Sciences Serie I. Mathematique 328 pp 117– (1999)
[21] Grozdena, Journal of Mathematical and Analytical Applications 239 pp 213– (1999) · Zbl 0934.35095
[22] Zhang, Nonlinear Analysis: Theory, Methods and Applications
[23] Garcia-Huidobro, Journal of Functional Analysis 172 pp 177– (2000)
[24] Coffman, Archive for Rational Mechanics and Analysis 46 pp 81– (1972)
[25] Kwong, Archive for Rational Mechanics and Analysis 105 pp 243– (1989)
[26] Chen, Communications in Partial Differential Equations 16 pp 1549– (1991)
[27] Mcleod, Transactions of the American Mathematical Society 339 pp 495– (1993)
[28] Segel, Annals of Mathematics 78 pp 339– (1963)
[29] Kato, Proceedings of Symposia in Pure Mathematics 45 pp 9– (1986) · doi:10.1090/pspum/045.2/843591
[30] Tsutsumi, Mathematica Japonica 17 pp 173– (1972)
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