  We consider asymptotically self-similar blow-up profiles of the thin film
equation consisting of a stabilising fourth order and destabilising second
order term. It has previously been shown that blow up is only possible when the
exponent in the second order term is above a certain critical value (dependent
on the exponent in the fourth order term). We show that in the limit that the
critical value is approached from above, the primary branch of similarity
profiles exhibits a well-defined structure consisting of a peak near the
origin, and a thin, algebraically decaying tail, connected by an inner region
equivalent (to leading order) to a generalised version of the Landau--Levich
`drag-out' problem in lubrication flow. Matching between the regions ultimately
gives the asymptotic relationship between a parameter representing the height
of the peak and the distance from the criticality threshold. The asymptotic
results are supported by numerical computations found using continuation.
