Well-posedness and Stability of Thermoelastic Suspension Bridges with Cattaneo Heat Conduction: The Role of Viscoelastic Memory

Abstract

Thermoelastic suspension bridges operate under strong thermo-mechanical coupling, and the classical Fourier law fails to capture the finite speed of thermal propagation observed in low-conductivity materials such as concrete. This paper investigates a suspension bridge model in which the heat flux is governed by the Cattaneo law with relaxation time $\tau$, and in which the bridge deck may or may not possess viscoelastic memory. Using operator semigroup theory, we establish well-posedness and derive sharp decay rates for both regimes. For the memoryless system ($\delta =0$), we show that the semigroup is polynomially stable with rate $t^{-1/2}$ when the structural parameters satisfy $\frac{{\rho }_1}{\kappa }=\frac{{\rho }_2}{b}$, and that the rate deteriorates to $t^{-1/4}$ otherwise; moreover, exponential stability fails whenever ${\chi }_0\ne 0$, or ${\chi }_0=0$ and ${\chi }_1=0$, where ${\chi }_0$ and ${\chi }_1$ are parameter-dependent coefficients. In sharp contrast, when viscoelastic memory is present ($\delta =1$), the semigroup becomes exponentially stable, regardless of the structural parameter ratio. This reveals a regularizing role of viscoelastic memory: it converts polynomial decay into exponential decay and removes the parameter-ratio obstruction. The results provide a theoretical basis for exploiting inherent viscoelasticity in long-span bridges and for optimizing deck parameters in memoryless structures.