  We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear
damped wave equation with nonlinear memory. Global existence and asymptotic
behavior as $t\rightarrow\infty$ of small data solutions have been established
in the case when $1\leq n\leq3.$ Moreover, we derive a blow-up result under
some positive data in any dimensional space.
