Multiple Quenching Solutions of a Fourth Order Parabolic PDE with a singular nonlinearity modelling a MEMS Capacitor

TL;DR

This paper investigates finite-time singularity formation in a fourth order nonlinear PDE modeling MEMS capacitors, revealing multiple quenching points and stable self-similar profiles through combined analytical and numerical methods.

Contribution

It introduces new numerical and analytical techniques to analyze multiple quenching solutions in a MEMS-related PDE with singular nonlinearity.

Findings

Singularity can form at multiple points in 1D and along a ring in 2D.

Asymptotic expansions accurately predict quenching locations.

Solutions converge to stable self-similar profiles at singularities.

Abstract

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric 2D case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.