Abstract: This paper is concerned with the initial value problem of a class of coupled nonlinear Klein-Gordon equations in two space dimensions.
The authors first establish the existence of the standing wave with the ground state by using variational calculus, next the authors derive out the sharp conditions for blowing-up and global existence in terms of the result, at last the authors show the instability of the standing wave with the ground state.

Key words: Nonlinear Klein-Gordon equations, Standing wave, Blow up, Instability,
Variational calculus, Ground states