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

Iqra Kanwal, Jianghao Hao, Muhammad Fahim Aslam, Zayd Hajjej, Mauricio Sepúlveda-Cortés, Rodrigo Vejár-Asem

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.

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

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.
Paper Structure (17 sections, 14 theorems, 148 equations, 4 figures, 1 table)

This paper contains 17 sections, 14 theorems, 148 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Well-Posedness
  4. Exponential Stability
  5. Blow Up
  6. Numerical Experiments
  7. Preliminaries
  8. Integration over time.
  9. Discretization of the bilaplacian.
  10. Treatment of the Fractional Derivative.
  11. Infinite memory term.
  12. Nonlinear term
  13. Assembly of the numerical scheme
  14. Computational results: exponential decay
  15. Computational results: blow-up in finite time
  16. ...and 2 more sections

Key Result

Lemma 2.1

Ferr1 If $0<\sigma<\frac{1}{2}$ and $f \in L^{2}(\Omega)$, then there is a unique $v \in H^2_{*}(\Omega)$ such that, for all $u \in H^2_{*}(\Omega)$, we have The function $u \in H^2_{*}(\Omega)$ satisfying eq2.1 is known as the weak solution to the stationary problem

Figures (4)

  • Figure 1: Initial displacement profile $u(x,y,0)$ of the bridge deck obtained from the truncated solution of the static problem $\Delta^2 u(x,y)=\sin(x)/10$.
  • Figure 2: Evolution of the discrete energy for different values of $p$, illustrating the decay regime.
  • Figure 3: Initial displacement profile $u(x,y,0)$ used in the blow-up experiment. The profile is obtained from the solution of the static problem $\Delta^2 u(x,y)=5\sin(x)$.
  • Figure 4: Time evolution of the numerical solution in the blow-up regime.

Theorems & Definitions (21)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 11 more