Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS

The singular parabolic problem $u_t=\mathrm{\Delta }u-\frac{\lambda f(x)}{(1+u{)}^2}$ on a bounded domain $\mathrm{\Omega }$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage --represented here by $\lambda$. As a byproduct, we prove that for sufficiently large $\lambda$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$.