Critical exponent for damped wave equations with nonlinear memory

We consider the Cauchy problem in Rn,n≥1Rn,n≥1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t→∞t→∞ of small data solutions have been established in the case when 1≤n≤31≤n≤3. We also derive a blow-up result under some positive data in any dimensional space.