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We prove that the thin film equation ht+div (hn grad (Δh))=0 in dimension d[ges ]2 has a unique C1 source-type radial self-similar non-negative solution if 0<n<3 and has no solution of this type if n[ges ]3. When 0<n3 the solution h has finite speed of propagation and we obtain the first order asymptotic behaviour of h at the interface or free boundary separating the regions where h>0 and h=0. (The case d=1 was already known [1]).

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We consider non-negative solutions on the half-line of the thin film equation h t + (h n h xxx) x = 0, which arises in lubrication models for thin viscous films, spreading droplets and Hele-Shaw cells. We present a discussion of the boundary conditions at x = 0 on the basis of formal and modelling arguments when x = 0 is an edge over which fluid can drain. We apply this discussion to define some similarity solutions of the first and the second kind. Depending on the boundary conditions, we introduce mass-preserving solutions of the first kind (0 < n < 3), 'anomalous dipoles' of the second kind (0 < n < 2, n = 1) and a standard dipole solution of the first kind for n = 1. For solutions of the first kind we prove results on existence, uniqueness and asymptotic behaviour, both at x = 0 and at the free boundary. For solutions of the second kind we briefly present some qualitative properties.