  We discuss a lubrication approximation model of the interface between two
immiscible fluids in a Hele-Shaw cell, derived in \cite{CDGKSZ93} and widely
studied since. The model consists of a single one dimensional evolution
equation for the thickness $2h = 2h(x,t)$ of a thin neck of fluid, \[
\partial_t h + \partial_x( h \, \partial_x^3 h) = 0\, , \] for $x\in (-1,1)$
and $t\ge 0$. The boundary conditions fix the neck height and the pressure
jump: \[ h(\pm 1,t) = 1, \qquad \partial_{x}^2 h(\pm 1,t) = P>0. \] We prove
that starting from smooth and positive $h$, as long as $h(x,t) >0$, for $x\in
[-1,1], \; t\in [0,T]$, no singularity can arise in the solution up to time
$T$. As a consequence, we prove for any $P>2$ and any smooth and positive
initial datum that the solution pinches off in either finite or infinite time,
i.e., $\inf_{[-1,1]\times[0,T_*)} h = 0$, for some $T_* \in (0,\infty]$. These
facts have been long anticipated on the basis of numerical and theoretical
studies.
