Dynamics of a Damped Wave Equation Arising from MEMS

Abstract

The semilinear, damped wave equation $\epsilon^2 u_{tt} + u_t - \Delta u = {-\lambda \over (1+u)^2}$, modeling an electrostatically actuated microelectromechanical system device, is studied. This equation describes the electrically driven motion of an elastic membrane oscillating above a ground plate. It is assumed that the motion starts from rest. The parameter $\lambda$ represents the applied voltage. It is known that there exists a critical value of $\lambda$, denoted by $\lambda_{*}$, such that there is at least one stationary solution for $\lambda < \lambda_{*}$ and none for $\lambda > \lambda_{*}$. It is also known that for the parabolic equation corresponding to $\epsilon=0$, this critical value represents the “pull-in” voltage. This pull-in voltage separates the stable operation regime for which the membrane approaches a steady state, from the “touchdown" regime for which the membrane collapses onto the plate. In the present work it is shown that $\lambda > \lambda_{*}$ belongs to the touchdown regime and that for large values of $\lambda$, touchdown takes place in finite time. In the one-dimensional spatial case, the existence of a stable operation regime is established for values of $\lambda$ smaller than a fixed number which is independent of $\epsilon$. Numerical solutions show that, unlike the parabolic case, the dynamical pull-in value is smaller than $\lambda_{*}$ and depends on $\epsilon$. The numerical evidence also shows that this dynamical critical value is a genuine threshold in the sense that it separates the stable operation regime from the touchdown regime.