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In the present paper, we study the Cauchy problem for the wave equation witha time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and acubic 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 thepresent paper is to prove a small data blow-up result and show an upperestimate 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 theconvolution term in high spatial dimensions, i.e. $n\ge 4$. This paper is thefirst blow-up result to treat wave equations with the cubic convolution in highspatial dimensions ($n\ge 4$).

Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data