Quenching for a semi-linear wave equation for MEMS

TL;DR

This paper investigates finite-time quenching singularities in semi-linear wave equations modeling MEMS, providing formal solutions, asymptotic analysis, and numerical evidence for the singular behavior.

Contribution

It introduces formal self-similar solutions and conjectures on singular behavior for MEMS-related wave equations with negative power nonlinearities.

Findings

Existence of quenching self-similar solutions

Radial solutions quench at the origin like explicit solutions

Numerical experiments support the conjectured singular behavior

Abstract

We consider the formation of finite-time quenching singularities for solutions of semi-linear wave equations with negative power nonlinearities, as can model micro-electro-mechanical systems (MEMS). For radial initial data we obtain, formally, the existence of a sequence of quenching self-similar solutions. Also from formal asymptotic analysis, a solution to the PDE which is radially symmetric and increases strictly monotonically with distance from the origin quenches at the origin like an explicit spatially independent solution. The latter analysis and numerical experiments suggest a detailed conjecture for the singular behaviour.