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In the present paper, we study the Cauchy problem for the wave equation witha time-dependent scale invariant damping $\frac{2}{1+t}\partial_t v$ and acubic convolution $(|x|^{-\gamma}*v^2)v$ with $\gamma\in\left(-\frac{1}{2},3\right)$ in three spatial dimension for initial data$\left(v(x,0),\partial_tv(x,0)\right)\in C^2(\mathbb{R}^3)\timesC^1(\mathbb{R}^3)$ with a compact support, where $v=v(x,t)$ is an unknownfunction to the problem on $\mathbb{R}^3\times[0,T)$. Here $T$ denotes amaximal existence time of $v$.  The first aim of the present paper is to prove unique global existence of thesolution to the problem and asymptotic behavior of the solution in thesupercritical case $\gamma\in (0,3)$, and show a lower estimate of the lifespanin the critical or subcritical case $\gamma\in \left(-\frac{1}{2},0\right]$.The essential part for their proofs is to derive a weaker estimate under theweaker condition than the case without damping and to recover the weakness bythe effect of the dissipative term.  The second aim of the present paper is to prove a small data blow-up and thealmost sharp upper estimate of the lifespan for positive data with a compactsupport in the subcritical case $\gamma\in \left(-\frac{1}{2},0\right)$. Theessential part for the proof is to refine the argument for the proof of Theorem6.1 in \cite{H20} to obtain the upper estimate of the lifespan.  Our two results determine that a critical exponent $\gamma_c$ which dividesglobal existence and blow-up for small solutions is $0$, namely $\gamma_c=0$.As the result, we can see that the critical exponent shift from $2$ to $0$ dueto the effect of the scale invariant damping term.

Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution