On the partial differential equations of electrostatic MEMS devices II: Dynamic case

Abstract

This paper is a continuation of [9], where we analyzed steady-states of the nonlinear parabolic problem $$u_{t} = \triangle u - \frac{\lambda f(x)}{(1+u)^{2}}$$ on a bounded domain Ω of $${\mathbb{R}}^{N}$$ with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at –1. Here u is modeled to describe dynamic deflection of the elastic membrane. When a voltage–represented here by λ– is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) may occur when it exceeds a certain critical value λ* (pull-in voltage), creating a so-called “pull-in instability” which greatly affects the design of many devices. In an effort to achieve better MEMS designs, the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f(x). We show that when $${\rm \lambda} \leq {\rm \lambda}^{*}$$ the membrane globally converges to its unique maximal steady-state. On the other hand, if λ > λ* the membrane must touchdown at finite time T, and that touchdown cannot occur at a location where the permittivity profile vanishes. We establish upper and lower bounds on first touchdown times, and we analyze their dependence on f, λ and Ω by applying various analytical and numerical techniques. A refined description of MEMS touchdown profiles is given in a companion paper [10].