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Stability and Finite-Time Blow-Up for a Fractionally Damped Nonlinear Plate Equation: Numerical and Analytical Insights

Iqra Kanwal, Jianghao Hao, Muhammad Fahim Aslam, Mauricio Sepúlveda-Cortés

TL;DR

This paper addresses the nonlinear plate equation with internal fractional damping and delayed velocity feedback, driven by a polynomial nonlinearity $|\,\mathcal{V}\,|^{p-2}\mathcal{V}$ ($p>2$). It develops an augmented memory-based formulation and proves local well-posedness via semigroup theory, followed by global existence and exponential energy decay under regime (A1). In the delay-dominated regime (A2) with negative initial energy, the concavity method yields finite-time blow-up. Numerical simulations using a beta-Newmark scheme with extended-variable discretization and Hermite cubic finite elements corroborate both stability and blow-up scenarios, highlighting the competing effects of fractional damping and delay.

Abstract

This paper studies a nonlinear plate equation with internal fractional damping and a time-delay term, driven by a polynomial-type nonlinear source. Such a model arises naturally in the description of viscoelastic and feedback-controlled elastic structures. We first establish the local existence and uniqueness of weak solutions using semigroup theory. The long-time behavior of solutions is then analyzed by constructing a suitable Lyapunov functional, from which stability and energy decay results are obtained. Moreover, by applying the concavity method, we prove that solutions associated with negative initial energy blow up in finite time. These results highlight the competing effects of fractional damping and delayed feedback on the qualitative behavior of the system. Finally, numerical simulations are presented to confirm the analytical results and to illustrate both stability and blow-up dynamics.

Stability and Finite-Time Blow-Up for a Fractionally Damped Nonlinear Plate Equation: Numerical and Analytical Insights

TL;DR

This paper addresses the nonlinear plate equation with internal fractional damping and delayed velocity feedback, driven by a polynomial nonlinearity (). It develops an augmented memory-based formulation and proves local well-posedness via semigroup theory, followed by global existence and exponential energy decay under regime (A1). In the delay-dominated regime (A2) with negative initial energy, the concavity method yields finite-time blow-up. Numerical simulations using a beta-Newmark scheme with extended-variable discretization and Hermite cubic finite elements corroborate both stability and blow-up scenarios, highlighting the competing effects of fractional damping and delay.

Abstract

This paper studies a nonlinear plate equation with internal fractional damping and a time-delay term, driven by a polynomial-type nonlinear source. Such a model arises naturally in the description of viscoelastic and feedback-controlled elastic structures. We first establish the local existence and uniqueness of weak solutions using semigroup theory. The long-time behavior of solutions is then analyzed by constructing a suitable Lyapunov functional, from which stability and energy decay results are obtained. Moreover, by applying the concavity method, we prove that solutions associated with negative initial energy blow up in finite time. These results highlight the competing effects of fractional damping and delayed feedback on the qualitative behavior of the system. Finally, numerical simulations are presented to confirm the analytical results and to illustrate both stability and blow-up dynamics.
Paper Structure (22 sections, 18 theorems, 159 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 18 theorems, 159 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Well-posedness
  4. Global existence
  5. Exponential Stability
  6. Blow-up Result
  7. Numerical Approximation
  8. Temporal discretization: The Newmark scheme with fractional and delay terms
  9. Discretization of the fractional derivative via extended variables
  10. Discretization of the delay term
  11. Complete discrete system
  12. Discrete energy decay
  13. Spatial discretization: Hermite cubic finite elements
  14. Mesh and basis functions
  15. Weak formulation
  16. ...and 7 more sections

Key Result

Lemma 2.1

MBO Let $\beta$ be the function that we will use to represent the damping effect for $\alpha \in \mathbb{R} \ \textit{on interval} \ 0<\theta<$1 and this function is defined as follows: Thus, the relationship between the 'input' $U$ and the 'output' $O$ of the system is described as follows: is given by where

Figures (5)

  • Figure 1: Exponential stability of the solution. Left: Evolution of displacement $\mathcal{V}(x,t)$. Right: Energy decay in semi-logarithmic scale.
  • Figure 2: Energy growth near the blow-up time $t\approx 0.04$.
  • Figure 3: Beam displacement shortly before blow-up. Left: full evolution until $t\approx 0.04$. Right: detailed view of the last $4\times10^{-4}$ seconds.
  • Figure 4: Violation of condition \ref{['eq4.1']}. Left: displacement evolution until blow-up ($t \approx 0.04$). Right: corresponding energy evolution.
  • Figure 5: Violation of condition (A1). Left: displacement evolution. Right: exponential energy growth without blow-up.

Theorems & Definitions (31)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • ...and 21 more