Abstract

Let A be a densely defined closed, linear ω-sectorial operator of angle θ∈[0,π2) on a Banach space X, for some ω∈ R. We give an explicit representation (in terms of some special functions) and study the precise asymptotic behavior as time goes to infinity of solutions to the following diffusion equation with memory: u′(t)=Au(t)+(κ∗Au)(t),t>0, u(0) = u, associated with the (possible) singular kernel κ(t)=αe-βttμ-1Γ(μ),t>0, where α∈ R, α≠ 0 , β≥ 0 and 0 < μ< 1.