Estimates for the Quenching Time of a Parabolic Equation Modeling Electrostatic MEMS

Abstract

The singular parabolic problem ut = ∆u − λf(x) (1+u)2 on a bounded domain Ω of R with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage —represented here by λ. As a byproduct, we prove that for sufficiently large λ, finitetime quenching must occur near the maximum point of the varying dielectric permittivity profile f(x).