Decay rate and eigenvalue asymptotics for magnetizable piezoelectric system with infinite memory

¹ School of Sports Economics and Management, Tianjin University of Sport, Tianjin 301617, China
² School of Mathematics and KL-AAGDM, Tianjin University, Tianjin 300350, China

^* Corresponding author: zjhan@tju.edu.cn

Received: 7 July 2023
Accepted: 3 June 2025

Abstract

In this paper, the asymptotic behavior of strongly coupled hyperbolic equations with one infinite memory term is investigated, one specific case of which is the model for describing the dynamical behavior of magnetic effected piezoelectric beams. A fractional operator is involved in the memory term depending on the parameter a ∈ [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 rate given as t^−1/2−2a is 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 spectrum for the system operator are derived. Based on the expressions of the spectrum, the optimality of the obtained decay rate is further verified for this system.