Finite time singularity in a free boundary problem modeling MEMS

An idealized electostatically actuated microelectromechanical system (MEMS) consists of a fixed horizontal ground plate held at zero potential above which an elastic membrane held at potential V is suspended. A Coulomb force is generated by the potential difference across the device and results in a deformation of the membrane, thereby converting electrostatic energy into mechanical energy, see [1, 4, 7] for a more detailed account and further references. After a suitable scaling and assuming homogeneity in transversal horizontal direction, the ground plate is assumed to be located at z = −1 and the membrane displacement u = u(t, x) ∈ (−1,∞) with t > 0 and x ∈ I := (−1, 1) evolves according to