Finite time singularity in a MEMS model revisited

TL;DR

This paper analyzes a MEMS model involving coupled equations for deformation and electrostatic potential, proving finite time singularities occur for all initial configurations when voltage exceeds a critical threshold, including snap-through instabilities.

Contribution

It extends previous results by proving finite time singularities for arbitrary initial conditions, not just non-positive ones, in a MEMS model.

Findings

Finite time singularity occurs when voltage exceeds critical value.

Singularities include snap-through instabilities.

Model validity breaks down at singularity.

Abstract

A free boundary problem modeling a microelectromechanical system (MEMS) consisting of a fixed ground plate and a deformable top plate is considered, the plates being held at different electrostatic potentials. It couples a second order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The validity of the model is expected to break down in finite time when the applied voltage exceeds a certain value, a finite time singularity occurring then. This result, already known for non-positive initial configurations of the top plate, is here proved for arbitrary ones and thus now includes, in particular, snap-through instabilities.