Optimal energy decay rate for a class of weakly dissipative second-order systems with memory

In this paper we study the asymptotic behavior of a class of linear dissipative integral differential equations u t t ( t ) + A u ( t ) − ∫ 0 ∞ g ( s ) A α u ( t − s ) d s = 0 . We show that the solution decays and the decay rate is optimal.