Asymptotic stability for the kirchhoff plate equation with a non linear delay term

Abstract

This study investigates the boundary value problem for a Kirchhoff-type plate equation, focusing on the qualitative properties and asymptotic behavior of solutions. The specificity of the considered model lies in the complex interplay between two competing mechanisms: a memory-based damping term that stabilizes the system by absorbing energy, and a nonlinear time delay that acts as a source of instability and perturbation amplification. We aim to establish the mathematical thresholds and constraints under which the damping forces prevail over the delay effects. This involves analyzing the impact of nonlinearities on the plate dynamics through the construction of appropriate Lyapunov functionals to precisely measure energy decay rates. Furthermore, we explore blow-up scenarios where the global existence of solutions fails, identifying the conditions that lead to system collapse due to the dominance of nonlinear sources.