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

ISSN： 1021-9722

Source： Nonlinear Differential Equations and Applications NoDEA, Vol.15, Iss.1-2, 2008-04, pp. : 115-145

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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].