On a hyperbolic equation arising in electrostatic MEMS

We consider the initial–boundary value problem of a damped wave equation with singular nonlinearity, which describes an electrostatic micro-electro-mechanical system (MEMS) device. The results of the pull-in voltage λ⁎λ⁎ being the critical threshold for global existence and quenching are obtained: if the applied voltage λ<λ⁎λ<λ⁎, then the equation admits a unique global small solution that exponentially converges to the minimal steady state, while large solution may quench in finite time; if λ>λ⁎λ>λ⁎, then any solution quenches in finite time. Finally, in the sense of the viscosity dominated limit, the asymptotic relation of solutions between the hyperbolic equation and the parabolic one is investigated. Also the related error estimates in arbitrary order are derived.