Distributed order derivatives and relaxation patterns

We consider equations of the form (D(ρ)u)(t)=-λu(t), t>0, where λ>0,D(ρ) is a distributed order derivative, that is D(ρ)φ(t)=∫01(D(α)φ)(t)dρ(α), where D(α) is the Caputo-Dzhrbashyan fractional derivative of order α, ρ is a positive measure. The above equation is used for modeling anomalous, non-exponential relaxation processes. In this work, we study the asymptotic behavior of solutions of the above equation, depending on the properties of the measure ρ.