Stability and Energy Decay for a Porous-Elastic System with One Global Fractional Damping

In this paper, we study the stability and the energy decay for a porous-elastic system with one global fractional damping. First, by the Lumer–Phillips theorem, we show that the system is well posed. Then, by the general criteria of Arendt–Batty, we show that the system is strongly stable. Finally, by the frequency-domain method, we show that the system is polynomially stable and that the corresponding energy decay rate depends on the wave speeds μρ,δJ and the parameters α,η in the Caputo fractional derivative operator. Specifically, if μρ=δJ and η>0, the optimal energy decay rate of the system is t−21−α; if μρ≠δJ and η>0, the optimal energy decay rate of the system is t−23−α; if μρ=δJ and η=0, the energy decay rate of the system is t−2; if μρ≠δJ and η=0, the energy decay rate of the system is t−23−α.