Thermal Relaxation Effects on Stability Thresholds: A Comparative Analysis of Thermoelastic Timoshenko-Boltzmann Systems Under Fourier and Cattaneo Laws

Abstract: This paper explores stability dichotomies of thermoelastic Timoshenko-Boltzmann systems with hereditary memory, comparing Fourier's parabolic (no thermal relaxation) and Cattaneo's hyperbolic (with thermal relaxation) heat conduction via semigroup theory and spectral analysis. The Fourier-semigroup achieves exponential stability through the $δ$-condition plus a specific parameter condition, while sole $δ$-condition ensures $1/2$-order polynomial stability. The Cattaneo-semigroup retains this polynomial stability but requires a thermal relaxation-included modified parameter condition for exponential stability. As thermal relaxation time approaches zero, Cattaneo's modified parameter converges to Fourier's, formalizing their asymptotic connection. Results show thermal relaxation alters exponential criteria but preserves polynomial decay, supporting stability analysis under time-dependent thermal loading in structural mechanics.

• The paper demonstrates that exponential stability in both Fourier and Cattaneo models is achieved when critical parameters vanish under precise conditions including the Chepyzhov–Pata memory kernel requirement.
• It employs an abstract semigroup approach to establish well-posedness and sharp resolvent estimates, clearly characterizing the shift between exponential and optimal polynomial decay.
• The findings indicate that while thermal relaxation modifies stability thresholds, the underlying mechanism for achieving robust polynomial decay remains intact.

Stability Thresholds in Thermoelastic Timoshenko-Boltzmann Systems with Thermal Relaxation

Introduction and Problem Formulation

This paper provides a rigorous comparative analysis of the asymptotic behavior and stability thresholds in thermoelastic Timoshenko-Boltzmann systems, focusing on the role of heat conduction laws—namely, the classical parabolic Fourier law and the hyperbolic Cattaneo law—when hereditary (Boltzmann-type) viscoelastic memory is present. The Timoshenko beam, accounting for both transverse displacement ($\phi$) and rotation ($\psi$), is coupled to a thermal field ($\theta$ and heat flux $q$), which responds via either instantaneous (Fourier, $\tau=0$) or relaxation-type (Cattaneo, $\tau>0$) thermal dynamics. The memory kernel $g$ induces viscoelastic dissipation, and $\sigma$ governs thermoelastic coupling. The central mathematical machinery is the abstract evolution equation in an appropriate Hilbert phase space, with generators $\mathcal{A}_F$ and $\mathcal{A}_C$ for Fourier and Cattaneo models, respectively.

The work aims to fill the gap in the literature concerning unification and comparison of stability properties when thermal relaxation and infinite memory coexist, generalizing and extending results on both the exponential and polynomial asymptotic stability of such systems.

Abstract Setting, Well-posedness, and Semigroup Generators

The paper sets up the coupled PDE-memory system as an abstract Cauchy problem in Hilbert spaces $\psi$0 (Fourier) and $\psi$1 (Cattaneo), built from displacement and temperature fields and the viscoelastic memory variable $\psi$2. The correspondence:

• $\psi$3 includes state variables $\psi$4.
• $\psi$5 augments this with the heat flux $\psi$6.

The memory variable is reformulated via translation semigroups, allowing an abstract semigroup approach. Well-posedness is established for both models: $\psi$7 and $\psi$8 are shown to be maximal dissipative (hence generate contraction $\psi$9-semigroups), under minimal assumptions on the kernel $\theta$0. Uniqueness and regularity of solutions for given initial data are established directly from the Lumer–Phillips theorem.

Spectral Analysis and Stability Thresholds

A sharp spectral analysis—invoking dissipativity, range conditions for the generator, and detailed estimates of the resolvent on the imaginary axis—leads to robust sufficient and necessary conditions for both (i) exponential and (ii) optimal polynomial decay. The core findings can be summarized as follows:

1. Exponential decay (strong stabilization):

• Fourier case ($\theta$1): Exponential stability holds if and only if:

(a) The memory kernel $\theta$2 satisfies the Chepyzhov–Pata $\theta$3-condition (a uniform exponential decay property for $\theta$4); (b) Critical parameter vanishes: $\theta$5 (synchronization of shear and bending wave speeds, the "equal wave speed" (EWS) condition).

• Cattaneo (thermal relaxation, $\theta$6): Exponential stability holds if and only if:

(a) $\theta$7 satisfies the $\theta$8-condition; (b) A modified parameter vanishes: $\theta$9. Thus, thermal relaxation shifts the stability threshold, introducing dependence on $q$0 and the coupling parameter $q$1.

2. Polynomial decay (weak stabilization):

• For both Fourier and Cattaneo models, as soon as the $q$2-condition (hence a degree of memory-induced dissipation) holds, but synchronization is lost, the semigroups are uniformly stable with the optimal $q$3-order polynomial rate:

$q$4

regardless of the values of $q$5 or $q$6. This reveals that the underlying dissipation mechanism from the hereditary memory is robust to both the specifics of wave synchrony and to thermal relaxation, at least for subexponential decay.

3. Asymptotic relationship:

As $q$7 (Cattaneo $q$8 Fourier), $q$9 and the generator $\tau=0$0, formalizing the continuous transition between relaxation and non-relaxation models at the level of spectral stability criteria.

Numerical and Analytical Sharpness

The results are substantiated by explicit resolvent estimates and construction of singular sequences, showing sharpness: exponential stability is lost as soon as the threshold is breached. The singular structures causing the loss of exponential decay are precisely characterized by sequences of eigenvalues and eigenvectors concentrating near critical parameter values.

Implications, Distinctions, and Future Directions

This analysis clarifies that, in the presence of both hereditary viscoelastic damping and thermal effects:

• Relaxational thermal effects (Cattaneo) fundamentally change the fine structure of the stability threshold, as opposed to the classical theory where only wave speed synchrony seems critical.
• However, the mechanism for optimal polynomial decay is robust—thermal relaxation does not weaken (but also does not improve) the slowest possible decay rates.
• For physical interpretations, this means that in applications (e.g., smart structures, advanced mechanical materials), the benefits of enhanced damping regimes (exponential, uniform decay) depend sensitively on precise parameter tunings, but the guarantee of polynomial decay persists widely.
• As time variation in boundary heat conditions or spatial inhomogeneity is incorporated, further weakening (to logarithmic or non-uniform decay) may occur—a topic for future work.

In operator-theoretic terms, the spectral shift induced by $\tau=0$1 in the critical condition can be seen as a nontrivial perturbation in the generator, and the limiting process (as $\tau=0$2) provides a concrete realization of singular perturbation theory in PDE-operator semigroups.

Further avenues include extension to nonlinear or time-dependent coefficients, non-smooth memory kernels, or multidimensional spatial domains, and considering more general thermomechanical couplings suggested in advanced continuum mechanics.

Conclusion

This paper rigorously establishes a parametric dichotomy for exponential versus polynomial stability in Timoshenko-type beam systems with infinite memory and thermal coupling. The main insight is that thermal relaxation—modeled via the Cattaneo law—not only modifies the threshold for exponential stability but also leaves the polynomial decay regime intact. The analytical tools and the explicit criteria derived here offer a unified framework to treat stabilization and convergence in various physically relevant settings for viscoelastic structures under complex thermal regimes, bridging the gap between parabolic (Fourier) and hyperbolic (Cattaneo) thermomechanics (2606.06956).