  In the present paper, we study the Cauchy problem for the wave equation with
a time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and a
cubic convolution $(|x|^{-\gamma}*v^2)v$ with $\gamma\in (0,n)$, where
$v=v(x,t)$ is an unknown function on $\mathbb{R}^n\times[0,T)$. Our aim of the
present paper is to prove a small data blow-up result and show an upper
estimate of lifespan of the problem for slowly decaying positive initial data
$(v(x,0),\partial_t v(x,0))$ such as $\partial_t v(x,0)=O(|x|^{-(1+\nu)})$ as
$|x|\rightarrow\infty$. Here $\nu$ belongs to the scaling supercritical case
$\nu<\frac{n-\gamma}{2}$. Our main new contribution is to estimate the
convolution term in high spatial dimensions, i.e. $n\ge 4$. This paper is the
first blow-up result to treat wave equations with the cubic convolution in high
spatial dimensions ($n\ge 4$).
