Dynamical Solutions of Singular Wave Equations Modeling Electrostatic MEMS
Yujin Guo · SIAM Journal on Applied Dynamical Systems · 2010
We study a fourth-order singular wave equation involving a singular nonlinear term ${\frac{\lambda}{(1-u)^2}}$ in a bounded domain of ${\mathbb{R}^N}$. This equation models a simple electrostatic microelectromechanical system (MEMS) device consisting of a thin elastic plate with boundary supported at 0 above a rigid ground plate located at 1. Here u is modeled to describe the dynamical deflection of the elastic plate. When a voltage—represented here by $\lambda$—is applied, the elastic plate deflects towards the ground plate, and snap-through (quenching) may occur when it exceeds a certain critical value $\lambda^*$ (the pull-in voltage), creating a so-called pull-in instability, which greatly affects the design of many devices. For $1\leq N\leq3$, analytic results show that there exist $0\lambda^*$ the elastic plate quenches at finite time.