The typical microelectromechanical system (MEMS) device is made of a rigid conducting ground plate above which a clamped deformable membrane coated with a thin conducting film is suspend. Application of a voltage difference induces a Coulomb force which, in turn, generates a displacement of the membrane. When the applied voltage exceeds a certain threshold value, the membrane might can lapse (touch down) on the ground plate.
The dynamic and stationary behavior of such MEM model is described by an initial-boundary value problem for the fourth order partial
differential equation with respect to the displacement \(u=u(t,x)\) of the membrane \(\Omega \subset \mathbb{R}\) \[ \gamma^{2} \partial^{2}_{t} u+\partial_{t}u+B\triangle^{2}u-T\triangle u=-\frac{\lambda}{(1+u)^2}, \quad t>0, x \in \Omega \]
\[ u=\partial_{v} u=0, t>0, x \in \partial\Omega; u(0,\cdot)=u^{0}, \partial_{t} u(0,\cdot)=u^{1}, x \in \Omega, \] where \(\gamma^{2} \partial^{2}_{t} u\) and \(\partial_{t} u\) account the inertia and damping effect, \(B \bigtriangleup^{2} u\) and \(-T\bigtriangleup u\) reflect the bending and stretching of the membrane, the right-hand-side of the PDE having the singularity \(u=-1\), reflects the action of the electrostatic force. The parameter \(\lambda\) is proportional to the square of the applied voltage. It is shown the existence of the threshold value \(\lambda_{*}>0\) such that no radially symmetric stationary solution exists for \(\lambda > \lambda_{*}\), but for \(\lambda \in (0,\lambda_{*})\) at least two solutions exists. For the relevant hyperbolic and parabolic evolution problems local and global well-posedness results are obtained together with the occurrence of finite time singularities when \(\lambda\geq \lambda_{*}\).