  The dynamical and stationary behaviors of a fourth-order evolution equation
with clamped boundary conditions and a singular nonlocal reaction term, which
is coupled to an elliptic free boundary problem on a non-smooth domain, are
investigated. The equation arises in the modeling of microelectromechanical
systems (MEMS) and includes two positive parameters $\lambda$ and $\varepsilon$
related to the applied voltage and the aspect ratio of the device,
respectively. Local and global well-posedness results are obtained for the
corresponding hyperbolic and parabolic evolution problems as well as a
criterion for global existence excluding the occurrence of finite time
singularities which are not physically relevant. Existence of a stable steady
state is shown for sufficiently small $\lambda$. Non-existence of steady states
is also established when $\varepsilon$ is small enough and $\lambda$ is large
enough (depending on $\varepsilon$).
