The time singular limit for a fourth-order damped wave equation for MEMS

TL;DR

This paper analyzes a coupled fourth-order damped wave equation modeling MEMS devices, focusing on the behavior of solutions as inertial effects diminish, and reviews recent existence and stability results.

Contribution

It investigates the time singular limit of the model when inertial effects vanish, providing insights into the touchdown behavior of the elastic plate in MEMS.

Findings

Analysis of solution behavior in the inertialless limit

Review of existence and non-existence of steady states

Insights into touchdown phenomena in MEMS

Abstract

We consider a free boundary problem modeling electrostatic microelectromechanical systems. The model consists of a fourth-order damped wave equation for the elastic plate displacement which is coupled to an elliptic equation for the electrostatic potential. We first review some recent results on existence and non-existence of steady-states as well as on local and global well-posedness of the dynamical problem, the main focus being on the possible touchdown behavior of the elastic plate. We then investigate the behavior of the solutions in the time singular limit when the ratio between inertial and damping effects tends to zero.