. In this paper, we present a mathematical analysis of a model representing a one-dimensional suspension bridge. The model incorporates the vibration of the bridge deck in the vertical plane, modeled as an extensible thermo elastic beam. The main cable is modeled as an elastic string connected to the roadbed through a distributed system of linear elastic springs. Heat conduction is governed by Cattaneo's law. We prove the existence and uniqueness of solutions using semigroup theory. We introduce a parameter X0, which reflects the interplay between elastic energy storage, inertial effects, and thermal dissipation, to serve as a criterion for stability. More precisely, we show that the associated semigroup is exponentially stable if and only if X0 = 0. Otherwise, we prove polynomial stability with an optimal decay rate. Finally, numerical approximations based on the finite element method for spatial discretization and the implicit Euler scheme for time integration are presented to validate the theoretical results.
. In the present paper, we study the Sine-Gordon equation with visco elastic memory damping and a non-zero external forcing. A primitive function of the memory kernel is assumed to be a generalized positive definite kernel (abbreviated as GPDK), which may oscillate or change sign. By combining the GPDK framework with the classical multiplier technique and an auxiliary system, we prove that the energy decays exponentially to a bounded residual (i.e., an ultimate bound) dictated by the external forcing and the domain volume. Numerical experiments quantitatively confirm these predictions, including the scaling of the bounded residual with the forcing amplitude and the kernel-controlled decay rate.
. Moving beyond traditional pullback attractors, this paper investigates the existence of random uniform attractors for a class of non-autonomous stochastic Klein-Gordon-Schr & ouml;dinger lattice dynamical systems. This approach effectively captures forward dynamics, overcoming the limitations of the pullback attractor. The system is driven by additive and multiplicative white noise and almost periodic nonlinearities; we rigorously prove the existence of the attractor within tempered closed bounded random sets.
The plate equation with linear memory and time-dependent perturbed parameter is mainly considered in a history space setting. Based on the established results concerning the time-dependent attractor and its asymptotic structure in [9], we verify the modified squeezing condition via the bootstrap method together with auxiliary norms defined for memory terms in time-dependent space, and thereby prove the existence of an exponential at-tractor along with higher regularity.
. The one-dimensional wave equation with boundary damping is exponentially stable, with a decay rate depending explicitly on the boundary feedback gain. Standard finite-difference and finite-element semi-discretizations, however, do not preserve this property uniformly as the mesh size tends to zero because of spurious high-frequency numerical modes. In this paper, we study Fourier-filtered semi-discrete approximations of the boundary-damped wave equation and establish uniform exponential stability for the retained modes. Our main contribution is a direct analysis of the filtered closed-loop semi-discrete systems themselves. More precisely, rather than relying solely on observability inequalities for the associated conservative discrete models, we construct discrete Lyapunov functionals for the damped finite-difference and finite-element approximations and derive explicit exponential decay estimates directly at the semi-discrete level. A key point in the analysis is the appearance of a higher-order discrete energy through the multiplier method; Fourier filtering provides the uniform spectral control needed to estimate this quantity in terms of the natural energy. For both the finite-difference and finite-element approximations, we obtain decay estimates whose rates depend explicitly on the feedback gain and the filtering threshold. In particular, the analysis makes precise how the filtering threshold enters the stability mechanism and shows that, after removal of the spurious high-frequency modes, the semi-discrete decay law recovers the structure of the continuous PDE estimate up to the unavoidable filtering factor. Numerical experiments illustrate the spectral behavior of the two approximations, the loss of uniform decay without filtering, and the improved stabilization of the filtered schemes.
