Manoel Jeremias dos Santos

We develop and analyze a fully discrete finite difference approximation for a damped shear beam model with mixed boundary conditions and no rotational inertia. The absence of a second‐order time derivative in the rotational equation gives rise to an asymmetric temporal structure. This structural feature motivates a discretization that is compatible with the spatial coupling of the two variables and preserves the underlying energy structure of the continuous problem. We introduce an implicit scheme on interlaced nodal and staggered grids and derive an exact discrete energy identity. The identity separates the physical damping from a nonnegative numerical dissipation generated by the implicit time stepping and yields unconditional stability in the discrete energy norm. Consistency is established with a truncation error of order , and convergence follows from stability and consistency by the Lax–Richtmyer equivalence theorem. We further derive a matrix representation that permits a recursive implementation of the coupled scheme. A manufactured‐solution test confirms the predicted first‐order convergence along the refinement path . Numerical experiments for the homogeneous problem illustrate the distinct long‐time behavior associated with the exponentially and polynomially stable regimes of the continuous model. Because the implicit discretization introduces additional dissipation, these experiments are interpreted qualitatively rather than as preservation of the continuous decay rates.

In this paper, we study the long-time behavior of solutions to the Shear beam model without rotational inertia, where a dissipative mechanism acts only on the equation governing the rotations of the cross-sections. Owing to the elliptic-hyperbolic structure of the system, we prove that the energy decays at a sharp polynomial rate. The optimality is established through resolvent estimates on the imaginary axis based on important arguments of [7]. This result contributes to the stabilization theory of partially damped beam models and settles an important open question concerning the asymptotic behavior in this class of systems.

In this work, we investigate the asymptotic behavior of solutions to a nonlinear model describing binary mixtures of solids. We establish the exponential decay of the energy in both the autonomous case (in the absence of external forces) and the non-autonomous case (under the influence of time-dependent external forces). For the autonomous system, we further analyze the observability properties of the associated dynamical system by providing a sufficient upper bound on the completeness defect that ensures a given set of bounded linear functionals constitutes a determining set. Numerical experiments are performed using the finite element method and Newmark’s algorithm to illustrate the conservative and dissipative behaviors of the proposed nonlinear coupled model. The results confirm the accuracy of the scheme and the exponential decay of the energy in dissipative cases.

This study examines the well-posedness and asymptotic behavior of a nonlinear suspension bridge with a deck modeled according to the Bresse vibration theory. We establish the existence and uniqueness of global solutions by means of semigroup theory of linear operators, which gives rise to a dynamical system of solutions. From the perspective of quasi-stability properties, we prove the existence of a global and exponential attractor for the dynamical system and the finiteness of the fractal dimension and the regularity of the attractor. Finally, we establish conditions for a finite set of bounded linear functionals defined on phase space to be a set of determining functionals for the dynamical system.