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

Content Provider	Springer Nature Link
Author	Ghoussoub, Nassif Guo, Yujin
Copyright Year	2008
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].
Ending Page	145
Page Count	31
Starting Page	115
File Format	PDF
ISSN	10219722
e-ISSN	14209004
Journal	Nonlinear Differential Equations and Applications NoDEA
Issue Number	1
Volume Number	15
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2008-05-22
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	touchdown points steady-state touchdown time global convergence Degenerate parabolic equations Analysis MEMS Nonlinear parabolic equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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Large time behavior of solutions to degenerate parabolic equations

Touchdown and related problems in electrostatic MEMS device equation

Homogenization of a nonlinear degenerate parabolic differential equation

Dynamics and touchdown in electrostatic MEMS

On the partial differential equations of electrostatic mems devices ii: dynamic case, in preparation.

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On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior, in preparation

On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior

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

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