Stability of Multi-dimensional Nonlinear Piezoelectric Beam with Viscoelastic Infinite Memory

TL;DR

This paper investigates the long-term stability of a nonlinear piezoelectric beam with viscoelastic infinite memory, demonstrating exponential decay of solutions through energy estimates and frequency domain analysis.

Contribution

It proves the exponential stability of a nonlinear coupled PDE system with only one viscoelastic memory term, extending stability results to nonlinear piezoelectric beams.

Findings

The coupled linear system can be stabilized exponentially with a single memory term.

The nonlinear system's solutions decay exponentially under certain conditions.

The well-posedness is established via semigroup and fixed point theories.

Abstract

The long time behavior of a kind of fully magnetic effected nonlinear piezoelectric beam with viscoelastic infinite memory is considered. The well-posedness of this nonlinear coupled PDEs system is showed by mean of the semigroup theories and Banach fixed point theorem. Based on frequency domain analysis, it is proved that the corresponding coupled linear system can be indirectly stabilized exponentially by only one viscoelastic infinite memory term, which is located on one equation of these strongly coupled PDEs. Then the exponential decay of the solution to the nonlinear coupled PDEs' system is established by the energy estimation method under certain condition.