Exponential stabilization and finite time blow-up in a fractional thermal piezoelectric beam with delay

This paper investigates the behavior of a nonlinear piezoelectric beam under electrostatic conditions, incorporating thermal effects, a tempered fractional memory term, and internal delays. A logarithmic source is introduced to model the strong nonlinear responses of materials, which goes beyond the capabilities of standard polynomial models. Key contributions of this study include the development of a unified framework that combines fractional damping, delay feedback, and thermoelastic coupling. We establish rigorous well-posedness results using semigroup theory and a fixed-point analysis. Additionally, we introduce a new Lyapunov functional that captures both thermal and hereditary dissipation, leading to sharp exponential decay estimates. Finally, we use a convexity argument to demonstrate that solutions with negative initial energy must blow up in finite time. These findings extend previous models of piezoelectric beams and offer novel insights into the stability and failure mechanisms of smart materials with complex damping and delay effects.