Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS

TL;DR

This paper analyzes the quenching time of an elastic membrane in an electrostatic MEMS model, providing estimates based on applied voltage and showing quenching occurs near maximum permittivity for large voltages.

Contribution

It offers new estimates for the quenching time of a parabolic PDE modeling MEMS and proves quenching occurs near the maximum of the dielectric profile for high voltages.

Findings

Quenching time estimates depend on applied voltage (x).

Finite-time quenching occurs near the maximum of (x) for large (x).

Abstract

The singular parabolic problem $u_t=\Delta u -\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $R^N$ 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 $\lambda$. As a byproduct, we prove that for sufficiently large $\lambda$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$.