Well-posedness and Stability Analysis of Suspension Bridge Models Coupled with Cattaneo Heat Conduction: The Role of Viscoelastic Memory

Technical Summary: Well-posedness and Stability Analysis of Suspension Bridge Models Coupled with Cattaneo Heat Conduction

Problem Formulation
This paper investigates the dynamic behavior and asymptotic stability of a coupled suspension bridge system modeled under the framework of the Timoshenko beam theory for the bridge deck and an elastic string for the main cable. A distinguishing feature of the proposed model is the incorporation of the Cattaneo heat conduction law, which accounts for thermal relaxation time and finite speed of heat propagation, contrasting with the instantaneous propagation assumed in classical Fourier laws. The system is governed by five coupled partial differential equations describing the vertical displacement of the main cable ($u$), the transverse displacement ($\phi$) and rotation angle ($\psi$) of the bridge deck, the temperature deviation ($\theta$), and the heat flux ($q$).

The study specifically analyzes two distinct scenarios controlled by a parameter $\delta$:

1. Memoryless Case ($\delta = 0$): The system lacks viscoelastic memory terms.
2. Memory Case ($\delta = 1$): The system includes a Boltzmann viscoelastic memory term acting on the bending stiffness of the bridge deck.

The primary objective is to establish the well-posedness of the system and determine the decay rates of the energy under these two conditions, paying particular attention to the regulatory role of viscoelastic memory in the presence of thermo-mechanical coupling.

Methodology
The authors employ Operator Semigroup Theory as the core mathematical framework. The system is reformulated as an abstract Cauchy problem $\frac{d}{dt}U = \mathcal{A}U$ in a suitable Hilbert space $\mathcal{H}$.

• Well-posedness: The existence and uniqueness of solutions are proven using the Lumer-Phillips Theorem. This involves demonstrating that the linear operator $\mathcal{A}$ is dissipative and maximal (i.e., $0$ belongs to the resolvent set). The proof utilizes the Lax-Milgram Theorem to establish the solvability of the associated elliptic boundary value problems.
• Stability Analysis:
  • For the memoryless case, polynomial stability is established using a frequency-domain approach based on the Gearhart-Prüss-Huang theorem (specifically Theorem 2.7 in the text). This involves deriving resolvent estimates for the operator $(i\lambda I - \mathcal{A})^{-1}$ as $|\lambda| \to \infty$. The analysis distinguishes between cases where the wave speeds of the bridge deck components are equal ($\rho_1/\kappa = \rho_2/b$) and unequal.
  • Non-exponential stability for the memoryless case is proven by constructing a specific sequence of eigenvalues and showing that the resolvent operator is not uniformly bounded, thereby violating the condition for exponential stability.
  • For the memory case, exponential stability is proven by combining the frequency-domain method (showing the imaginary axis is in the resolvent set) with energy estimates involving the memory kernel $g(s)$. The proof relies on the assumption that the kernel decays exponentially ($g'(s) \leq -\kappa_1 g(s)$).

Key Contributions and Results

1. Well-posedness: The paper rigorously proves that both the memoryless ($\delta=0$) and memory-inclusive ($\delta=1$) systems generate $C_0$-contraction semigroups, ensuring the existence and uniqueness of mild and classical solutions for appropriate initial data.

2. Polynomial Stability in the Memoryless Case ($\delta=0$):

   • The system exhibits polynomial decay rather than exponential decay.
   • The decay rate is contingent on the relationship between structural parameters:
     • If $\frac{\rho_1}{\kappa} = \frac{\rho_2}{b}$, the energy decays at a rate of $t^{-1/2}$.
     • If $\frac{\rho_1}{\kappa} \neq \frac{\rho_2}{b}$, the decay rate slows to $t^{-1/4}$.
   • The authors identify specific coefficient conditions ($\chi_0 \neq 0$ or $\chi_0 = 0, \chi_1 = 0$) under which exponential stability is strictly unachievable.

3. Exponential Stability in the Memory Case ($\delta=1$):

   • The introduction of the viscoelastic memory term fundamentally alters the stability profile. The system achieves exponential decay of energy.
   • This result holds regardless of the specific relationship between the structural parameters ($\rho_1/\kappa$ and $\rho_2/b$), indicating that viscoelastic memory acts as a dominant damping mechanism that overcomes the limitations of thermoelastic dissipation alone.

4. Role of Cattaneo Law: The study highlights that the Cattaneo heat conduction law introduces a thermal delay (relaxation time $\tau$). In the absence of viscoelastic memory, this delay, combined with the specific coupling coefficients, contributes to the polynomial rather than exponential decay, a finding consistent with but refined from previous Fourier-based models.

Significance and Claims
The paper claims to address critical gaps in existing literature regarding suspension bridge modeling:

• Thermal Modeling: By adopting the Cattaneo law, the model more accurately reflects the physical reality of thermal propagation in materials like concrete, which exhibit time delays not captured by Fourier's law.
• Unified Viscoelastic Analysis: The adjustable parameter $\delta$ allows for a unified theoretical treatment of both memoryless and memory-dependent scenarios, filling a gap where previous studies either ignored memory entirely or treated it as a fixed feature.
• Comprehensive Coupling: The model integrates four core mechanisms: main cable vibration, Timoshenko beam dynamics (transverse and rotational), Cattaneo-type heat conduction, and thermo-mechanical coupling, providing a more holistic framework than simplified prior models.

The authors conclude that viscoelastic memory is a crucial factor for enhancing structural stability. While memoryless systems rely on specific parameter tuning to achieve optimal polynomial decay, the inclusion of memory ensures robust exponential stability, offering valuable theoretical guidance for the design and safety assessment of suspension bridges, particularly those utilizing materials with significant viscoelastic properties.