Stability and blow-up for a suspension bridge plate model with fractional damping and memory
This work addresses stability and failure dynamics in suspension bridges, which is incremental as it extends existing models with fractional damping and memory effects.
The paper tackles a suspension bridge model with fractional damping and memory, establishing conditions for global stability and finite-time blow-up of solutions, and validates these regimes through numerical experiments.
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.