  We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon
equations in one space dimension. Under a suitable structural condition on the
nonlinearity, we will show that the solution exists globally and decays of the
order $O(t^{-1/2})$ in $L^\infty$ as $t$ tends to infinity without the
condition of a compact support on the Cauchy data which was assumed in the
previous works.
