Quenching for a semi-linear wave equation for micro-electro-mechanical systems

Abstract

We consider the formation of finite-time quenching singularities for solutions of semi-linear wave equations with negative power nonlinearities, such as can model micro-electro-mechanical systems. For radial initial data we obtain, formally, the existence of a sequence of quenching self-similar solutions. Also from formal aymptotic 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.