Free vibrations in a wave equation modeling MEMS

Abstract

We study a nonlinear wave equation appearing as a model for a membrane (without viscous effects) under the presence of an electrostatic potential with strength $\lambda$. The membrane has a unique stable branch of steady states $u_{\lambda }$ for $\lambda \in [0,{\lambda }_{\ast }]$. We prove that the branch $u_{\lambda }$ has an infinite number of branches of periodic solutions (free vibrations) bifurcating when the parameter $\lambda$ is varied. Furthermore, using a functional setting, we compute numerically the branch $u_{\lambda }$ and their branches of periodic solutions. This approach is useful to validate rigorously the steady states $u_{\lambda }$ at the critical value ${\lambda }_{\ast }$.