Asymptotics of selfsimilar blowup profiles of the thin film equation

We consider asymptotically self-similar blow-up profiles of the thin film equation. It has previously been shown that blow up is only possible when the exponents in the thin film equation are above a certain criticality threshold. We show that in the limit that the criticality threshold is approached from above, the similarity profiles exhibit a well-defined structure consisting of a peak near the origin, and a thin algebraically decaying tail; these are connected by an inner region, which is mathematically (to leading order) equivalent to the problem near an apparent contact line in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between the height of the peak and the distance from the criticality threshold, from which all other properties of the profile may be deduced.