Asymmetric fractional coupled magnetizable piezoelectric beams with infinite viscoelastic memory: polynomial decay and sharpness of the decay rate

Abstract

This paper studies the long-time dynamics of a coupled hyperbolic system for magnetizable piezoelectric beams with infinite viscoelastic memory. Memory dissipation is characterized by $A^{\hat{I}}\pm$, the fractional power of a positive self-adjoint operator $A$ with $\hat{I}\pm \in [0,1)$. An asymmetric fractional magnetoelectric coupling is adopted: the mechanical-to-magnetic coupling uses integer-order operator $A$, while the magnetic feedback to mechanics is governed by fractional operator $A^2$ (${\hat{I}}^2\in [0,1)$). This model bridges the gap between the well-studied integer coupling case (${\hat{I}}^2=1$) and the unsolved fully fractional symmetric coupling problem, offering a universal framework for related coupled systems. Under mild assumptions on memory kernels and system parameters, we prove well-posedness via semigroup theory and derive an explicit polynomial decay estimate for smooth initial data: \[ \|X(t)\|_{\mathcal H} \le C t^{-\frac{1}{4-2β-2α}}\|X_0\|_{D(\mathcal A)},\quad \forall\,t\ge 1, \] where the decay exponent is explicitly determined by $\hat{I}\pm$ and ${\hat{I}}^2$. For exponentially decaying memory kernels, the decay rate is sharp if stiffness coefficients satisfy $\hat{I}{\pm }_1\ne \hat{I}{\pm }_2$. When $\hat{I}{\pm }_1=\hat{I}{\pm }_2$, we only obtain an upper bound $\hat{I}´\le \frac{1}{3-{\hat{I}}^2-2\hat{I}\pm }$ for decay index $\hat{I}´$, leaving the optimal rate open. Comparisons with integer feedback coupling (${\hat{I}}^2=1$) show fractional feedback (${\hat{I}}^2<1$) slows energy decay. It demonstrates that ${\hat{I}}^2$ weakens indirect damping and degrades structural stabilization. The results reveal the intrinsic interaction between fractional memory dissipation and fractional coupling in strongly coupled dissipative systems.