  We consider an integro-differential counterpart of the $\sigma-$evolution
equation of the type \[ \partial_t^2 u(t,x)+\mu (-\Delta)^{\frac{\sigma}{2}}
\partial_t u(t,x)+(-\Delta)^\sigma u(t,x)=f(t,x), \] with $\sigma>0$ and
$\mu>0$, that encodes memory of \textit{power-law} type. To do so, we replace
the time derivatives $\partial_t$ and $\partial_t^2$ by the so-called
Caputo-Djrbashian derivatives $\partial_t^\gamma$ of order $\gamma=\alpha$ and
$\gamma=2\alpha$, respectively, and the inhomogeneous term $f(t,x)$ by the
Riemann-Liouville integral $I^{\beta-2\alpha}_{0^+}f(t,x)$, whereby
$0<\alpha\leq 1$ and $2\alpha\leq \beta<2\alpha+1$. For the solution
representation of the underlying Cauchy problems on the space-time $[0,T]\times
\mathbb{R}^n$ we then consider a wide class of pseudo-differential operators
$\displaystyle
(-\Delta)^{\frac{\eta}{2}}E_{\alpha,\beta}\left(~-\lambda(-\Delta)^{\frac{\sigma}{2}}
t^\alpha~\right)$, endowed by the fractional Laplacian
$-(-\Delta)^{\frac{\sigma}{2}}$ and the two-parameter Mittag-Leffler functions
$E_{\alpha,\beta}$. On our approach we are also able to provide dispersive and
Strichartz estimates for the solutions with the aid of decay properties of
$E_{\alpha,\beta}(-z)$ ($z\in \mathbb{C}$) and the boundedness properties of
the Hankel transform.
