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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.

On the critical decay for the wave equation with a cubic convolution in 3D