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

Technical Summary: Asymmetric Fractional Coupled Magnetizable Piezoelectric Beams with Infinite Viscoelastic Memory

Problem Statement
This paper investigates the long-time dynamics and stability of a coupled hyperbolic system modeling magnetizable piezoelectric beams subject to infinite viscoelastic memory. The study addresses a specific gap in the mathematical modeling of such systems: the interaction between fractional-order viscoelastic memory and fractional-order magnetoelectric coupling.

While previous works have established stability for systems with integer-order coupling (where the coupling operator is the standard Laplacian or its equivalent, $A$) or for fully symmetric fractional systems (where both coupling directions use fractional powers), the general case involving asymmetric fractional coupling remained unresolved. Specifically, the paper targets the abstract system defined by:
$$\begin{cases} v_{tt}(t) = -\alpha_1 Av(t) + kAp(t) + \int_0^\infty g(s)A^\alpha v(t-s)ds, \\ p_{tt}(t) = -\alpha_2 Ap(t) + kA^\beta v(t), \end{cases}$$
where $v$ represents mechanical displacement and $p$ represents magnetic potential. The system features:

• Infinite Memory: A viscoelastic term acting on the mechanical equation via the fractional operator $A^\alpha$ ($\alpha \in [0,1)$).
• Asymmetric Coupling: The mechanical-to-magnetic coupling uses the integer-order operator $A$ (effectively $\beta=1$ in that direction), while the magnetic-to-mechanical feedback uses a fractional operator $A^\beta$ ($\beta \in [0,1)$).

The core objective is to determine the precise polynomial decay rate of the system's energy and to establish whether this rate is sharp, particularly analyzing how the fractional orders $\alpha$ and $\beta$ jointly influence the dissipation mechanism.

Methodology
The authors employ a functional analytic approach based on semigroup theory and frequency-domain analysis:

1. Well-posedness via Semigroup Theory:

   • The non-autonomous system with memory is reformulated as an autonomous evolution equation in an extended Hilbert space $\mathcal{H}$. This space incorporates the state variables $(v, v_t, p, p_t)$ and a history variable $\eta(t,s) = v(t) - v(t-s)$.
   • A specific inner product is defined on $\mathcal{H}$ to ensure the energy norm is equivalent to the standard norm, relying on parameter assumptions (A2) regarding stiffness coefficients ($\alpha_1, \alpha_2$) and coupling strength ($k$).
   • Using the Lumer-Phillips theorem, the authors prove that the associated linear operator generates a contractive $C_0$-semigroup, establishing the existence and uniqueness of mild and classical solutions.

2. Polynomial Stability (Frequency-Domain Analysis):

   • To derive decay rates, the paper utilizes the Borichev-Tomilov characterization, which links polynomial decay to the growth of the resolvent operator $(i\lambda I - \mathcal{A})^{-1}$ along the imaginary axis.
   • The authors perform a detailed spectral analysis, assuming the contrary that the resolvent grows too fast. By constructing a sequence of approximate solutions and utilizing multiplier techniques, they derive contradiction estimates.
   • Key steps involve estimating the norms of the memory term, the velocity, and the displacement in terms of the spectral parameter $\lambda$, carefully balancing the fractional powers $\alpha$ and $\beta$.

3. Sharpness Analysis (Spectral Asymptotics):

   • To verify if the derived decay rate is optimal, the authors analyze the asymptotic behavior of the eigenvalues of the system operator for specific memory kernels (exponentially decaying).
   • They construct specific test functions based on the eigenfunctions of the operator $A$ to demonstrate that the resolvent norm indeed grows at the rate predicted by the decay exponent, confirming sharpness under certain stiffness conditions.

Key Contributions and Results

• Well-posedness: The paper rigorously proves that the asymmetric fractional coupled system is well-posed in the extended Hilbert space $\mathcal{H}$ under mild assumptions on the memory kernel and system parameters.
• Explicit Polynomial Decay Rate: For smooth initial data, the authors derive an explicit polynomial decay estimate:
  $$\|X(t)\|_{\mathcal{H}} \leq C t^{-\frac{1}{4-2\beta-2\alpha}} \|X_0\|_{D(\mathcal{A})}, \quad \forall t \geq 1.$$
  This result explicitly quantifies the decay exponent as a function of both the memory fractional order $\alpha$ and the magnetic feedback fractional order $\beta$.
• Comparison with Integer Coupling: The analysis reveals that introducing fractional feedback coupling ($\beta < 1$) slows down the energy decay compared to the integer coupling case ($\beta = 1$). Specifically, the denominator in the decay exponent increases as $\beta$ decreases, indicating that fractional feedback weakens the indirect damping effect transmitted from the mechanical to the magnetic component.
• Sharpness and Optimality:
  • Case $\alpha_1 \neq \alpha_2$: If the stiffness coefficients are distinct, the derived decay rate $t^{-\frac{1}{4-2\beta-2\alpha}}$ is proven to be sharp.
  • Case $\alpha_1 = \alpha_2$: When stiffness coefficients are identical, the authors' analysis yields an upper bound for the decay index $\delta \leq \frac{1}{3-\beta-2\alpha}$. This suggests a potential loss of sharpness in the symmetric case, or that the current technical tools are insufficient to confirm the optimality of the $4-2\beta-2\alpha$ exponent in this specific configuration.

Significance and Claims
The paper claims to bridge the theoretical gap between the well-studied integer-coupling case and the unsolved fully fractional symmetric coupling problem. By introducing an asymmetric fractional framework, the authors provide a universal model that allows for the separate quantification of how fractional feedback ($A^\beta$) modulates energy decay.

The primary significance lies in:

1. Generalization: Extending the decay rate formula to include the fractional coupling order $\beta$, thereby generalizing previous results that were limited to $\beta=1$.
2. Mechanism Insight: Demonstrating that the fractional order $\beta$ directly weakens the indirect damping mechanism, thereby degrading the overall structural stabilization performance.
3. Nuance in Symmetry: Uncovering a subtle distinction in optimality based on the equality of stiffness coefficients ($\alpha_1 = \alpha_2$ vs. $\alpha_1 \neq \alpha_2$), a phenomenon not observed in prior integer-coupling studies.

The authors modestly note that the exact optimal decay rate for the symmetric case ($\alpha_1 = \alpha_2$) remains an open problem, leaving the question of whether the true rate is governed by the exponent $1/(3-\beta-2\alpha)$ or $1/(4-2\beta-2\alpha)$ for future investigation.