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

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.