Dipoles and similarity solutions of the thin film equation in the half - line ∗

We consider non-negative solutions on the half-line of the thin film equation ht + (hhxxx)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 6= 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. AMS classification scheme numbers: 35K55, 35K65, 76D08, 34C35