Stability and blow-up for a suspension bridge plate model with fractional damping and memory

TL;DR

This paper analyzes a nonlinear suspension bridge model with fractional damping and memory, establishing conditions for stability, blow-up, and providing numerical simulations that confirm the theoretical results.

Contribution

It introduces a comprehensive analysis of a fractional damping suspension bridge model, including well-posedness, stability, blow-up conditions, and a structure-preserving numerical scheme.

Findings

Global exponential stability under certain conditions

Finite-time blow-up for negative initial energy

Numerical simulations confirm theoretical stability and blow-up regimes

Abstract

We investigate a suspension bridge model described by a nonlinear plate equation incorporating internal fractional damping and infinite memory effects. The system also includes a nonlinear source term that may induce instability. Using semigroup theory, we first establish the local well-posedness of solutions in an appropriate energy space. We then derive conditions ensuring global existence and exponential stability of solutions. In contrast, when the initial energy is negative, we prove that solutions blow up in finite time, revealing a threshold phenomenon governing the long-term dynamics of the system. To complement the analytical results, we construct a numerical approximation using Summation-By-Parts finite differences with Simultaneous Approximation Terms (SBP-SAT) for spatial discretization and a Newmark scheme for time integration. The scheme preserves the structural properties of the continuous energy framework. Numerical experiments illustrate the stability and blow-up regimes predicted by the theoretical analysis.