The focus of the current paper is to prove nonexistence results for the following Cauchy problem of a wave equation with fractional damping and non-linear memory
\begin{equation}
u_{tt} – \Delta u + D_{0|t}^{\sigma}u_{t} = \int_{0}^{t} (t – \tau)^{-\gamma} |u(\tau, \cdot)|^{p} d\tau, \tag{1}
\end{equation}
\begin{equation}
u(0,x) = u_{0}(x), \quad u_{t}(0,x) = u_{1}(x), \qquad x \in \mathbb{R}^{N}, \tag{2}
\end{equation}
where $p > 1$, $0 < \gamma < 1$ and $\Delta$ is the usual Laplace operator, $\sigma \in ]0,1[$ and $D_{0|t}^{\sigma}$ is the right-hand side fractional operator of Riemann–Liouville. Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions.