IMA Journal of Applied Mathematics Advance Access published online on March 25, 2009
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxp004
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Non-existence of global solutions of a class of coupled non-linear Klein–Gordon equations with non-negative potentials and arbitrary initial energy
Institute of Applied Physics and Computational Mathematics, PO Box 8009-15, Beijing 100088, China
Email: wang_jasonyj2002{at}yahoo.com, wang_yanjin{at}iapcm.ac.cn.
Received on November 28, 2007; Revision received April 23, 2008. Accepted on January 18, 2009
In the paper, we considerthe non-existence of global solutions of Cauchy problem for coupled Klein–Gordon equations of the form
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x n. First, for the case n = 2, 3, we prove the existence of ground state of the corresponding Lagrange–Euler equations of the above equations. Then, we establish a blow-up result with low initial energy, which leads to instability of standing waves of the system above. Moreover, as a byproduct we also discuss the global existence. Next, based on concavity method, we prove the blow-up result for the system with non-positive initial energy in the general case: 1 
n < 6. Finally, when the initial energy is given arbitrarily positive, we show that if the initial datum satisfies some conditions, the corresponding solution blows up in a finite time. In other words, in this paper we establish the complete blow-up result for the Klein–Gordon equation above in the sense of the initial energy, – 
< E(0) < + .
Keywords: coupled Klein–Gordon equations; variational calculus; blow-up; arbitrarily initial energy; non-negative potential.