ANALYSIS OF THE DYNAMICS AND TOUCHDOWN IN A MODEL OF ELECTROSTATIC MEMS

We study a reaction-diffusion equation in a bounded domain in the plane, which is a mathematical model of an idealized electrostatically actuated microelectromechanical system (MEMS). A relevant feature in these systems is the "pull-in" or "jump-to contact" instability, which arises when applied voltages are increased beyond a critical value. In this situation, there is no longer a steady state configuration of the device where mechanical members of the device remain separate. It may present a limitation on the stable operation regime, as with a micropump, or it may be used to create contact, as with a microvalve. The applied voltage appears in the equation as a parameter. We prove that this parameter controls the dynamics in the sense that before a critical value the solution evolves to a steady state configuration, while for larger values of the parameter, the "pull-in" instability or "touchdown" appears. We estimate the touchdown time. In one dimension, we prove that the touchdown is self-similar and determine the asymptotic rate of touchdown. The same type of results are obtained in a disk. We also present numerical simulations in some two-dimensional domains which allow an estimate of the critical voltage and of the touchdown time. This information is relevant in the design of the devices.