Decay Rates and Eigenvalue Asymptotics for Abstract Strongly Coupled Hyperbolic Equations with Infinite Memory

In this paper, the asymptotic behavior of abstract strongly coupled hyperbolic equations with one infinite memory term is investigated, one specific case of which is the model for describing the dynamical behaviour of magnetic effected piezoelectric beams. A fractional operator is involved in the memory term depending on the parameter $a\in [0,1)$. By means of frequency domain analysis, it is proved that the system can be indirectly stabilized polynomially by only one infinite memory term located on one of these two strongly coupled PDEs, and the explicit decay rates given as $t^{-\frac{1}{2-2a}}$ are only dependent on the parameter $a$. When considering the exponentially decreasing kernel, a detailed spectral analysis for the system operator is further provided. Specifically, the asymptotic expressions of the eigenvalues of the system operator are derived. Based on the expressions of the eigenvalues, the optimality of the obtained decay rates is further verified for this system.