Stability for Abstract Coupled Second-Order Evolution Equations with One Fractional Infinite Memory

In this paper, we study the stability for abstract coupled second-order evolution equations with one fractional infinite memory. By the frequency domain method, we show that the stability of the system depends on μϱ,aJ, and the order θ of the fractional operator, where θ∈[0,1]. Specifically, the system is exponentially stable for θ=1 and μϱ=aJ. The system is polynomially stable for μϱ=aJ and θ∈[0,1) and the corresponding energy decay rate is t−11−θ. The system is polynomially stable for μϱ≠aJ and θ∈[0,1] and the corresponding energy decay rate is t−12−θ. Moreover, we show that the polynomial decay rates obtained in some sense are optimal.