  We consider the wave equation with a cubic convolution $\partial_t^2 u-\Delta
u=(|x|^{-\gamma}*u^2)u$ in three space dimensions. Here, $0<\gamma<3$ and $*$
stands for the convolution in the space variables. It is well known that if
initial data are smooth, small and compactly supported, then $\gamma\ge2$
assures unique global existence of solutions. On the other hand, it is also
well known that solutions blow up in finite time for initial data whose decay
rate is not rapid enough even when $2\le \gamma<3$. In this paper, we consider
the Cauchy problem for $2\le \gamma<3$ in the space-time weighted $L^\infty$
space in which functions have critical decay rate. When $\gamma=2$, we give an
optimal estimate of the lifespan. This gives an affirmative answer to the Kubo
conjecture (see Remark right after Theorem 2.1 in Kubo(2004)). When
$2<\gamma<3$, we also prove unique global existence of solutions for small
data.
