Energy Decay Rate for a Porous-Elastic System with Fractional Infinite Memory

In this paper, we study the energy decay rate for a porous-elastic system with fractional infinite memory, which acts on the material displacement equation. Using the frequency domain method, we show that the energy decay rate of the system depends on the order θ of the fractional operator and the wave speeds μρ and δJ, where θ∈[0,1]. Specifically, the energy of the system decays exponentially when θ=1 and μρ=δJ. The energy of the system decays polynomially when θ∈[0,1) and μρ=δJ or θ∈[0,1] and μρ≠δJ. In addition, we also show that the polynomial decay rates obtained are optimal in some sense.