On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS control

TL;DR

This paper studies a nonlocal parabolic model for electrostatic MEMS, demonstrating that solutions can quench at a single point without steady states, with numerical simulations supporting the theoretical findings.

Contribution

It establishes conditions for solution quenching in a nonlocal MEMS model and provides bounds and numerical illustrations of touchdown behavior.

Findings

Solution quenches at a single point

Touchdown occurs without steady states

Numerical simulations confirm theoretical results

Abstract

We consider a nonlocal parabolic model for a micro-electro-mechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a situation where there is no steady state. It is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained. Numerical simulations illustrating the results are given.