  In the paper we consider the nonexistence of global solutions of the Cauchy
problem for coupled Klein-Gordon equations of the form \begin{eqnarray*}
\left\{\begin{array}{l} u_{tt}-\Delta u+m_1^2 u+K_1(x)u=a_1|v|^{q+1}|u|^{p-1}u
v_{tt}-\Delta v+m_2^2 u+K_2(x)v=a_2|u|^{p+1}|v|^{q-1}v u(0,x)=u_0;
u_t(0,x)=u_1(x) v(0,x)=v_0; v_t(0,x)=v_1(x) \end{array} \right. \end{eqnarray*}
on $\R\times\R^n$.
  Firstly for some special cases of $n=2,3$, 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: $n\geq 1$. 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.
