On a free boundary model for three-dimensional MEMS with a hinged top plate II: Parabolic case
Katerina Nik · Communications on Pure & Applied Analysis · 2021
A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter \begin{document}$ \lambda $\end{document} which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed in time and that, for \begin{document}$ \lambda $\end{document} sufficiently small, solutions exist globally in time. In addition, touchdown of the top plate on the ground plate is shown to be the only possible finite time singularity.