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

Abstract

This paper focuses on investigating a class of suspension bridge models coupled with the Cattaneo heat conduction law, with special attention paid to two distinct scenarios: systems with viscoelastic memory and those without. Operator semigroup theory is adopted as the core mathematical tool to conduct a comprehensive analysis of the models' dynamic behaviors. For the memoryless suspension bridge system ($\delta = 0$), we first establish its well-posedness (i.e., the existence and uniqueness of solutions). Further stability analysis reveals that this system exhibits polynomial decay behavior, where the decay rate depends on the structural parameter relationship of the bridge deck: a decay rate of $t^{-1/2}$ is achieved when $\frac{\rho_1}{\kappa} = \frac{\rho_2}{b}$, and the rate slows down to $t^{-1/4}$ when $\frac{\rho_1}{\kappa} \neq \frac{\rho_2}{b}$. Moreover, exponential stability is proven to be unachievable under specific coefficient configurations (i.e., $\chi_0 \neq 0$, or $\chi_0 = 0$ and $\chi_1 = 0$, with $\chi_0, \chi_1$ being parameter-dependent coefficients). For the counterpart system incorporating viscoelastic memory ($\delta = 1$), the well-posedness of solutions is also verified, and more importantly, the system is shown to achieve exponential decay. This indicates that viscoelastic memory significantly enhances system stability and accelerates energy dissipation compared to the memoryless case. By systematically exploring the regulatory role of viscoelastic memory in the stability of suspension bridge systems under the framework of Cattaneo thermal conduction, this study enriches and extends the existing research on suspension bridge models with viscoelastic memory.