  We consider a hyperbolic ordinary differential equation perturbed by a
nonlinearity which can be singular at a point and in particular this includes
MEMS type equations. We first study qualitative properties of the solution to
the stationary problem. Then, for small value of the perturbation parameter as
well as initial value, we establish the existence of a global solution by means
of the Lyapunov function and we show that the omega limit set consists of a
solution to the stationary problem. For strong perturbation or large initial
value, we show that the solution blows up. Finally, we discuss the relationship
between upper bounds of the perturbation parameter for the existence of
time-dependent and stationary solutions, for which we establish an optimal
threshold.
