| dc.description |
We consider non-negative solutions on the half-line of the thin film equation h<sub>t</sub> +(h<sup>n</sup> h<sub>xxx</sub> )<sub>x </sub> = 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. |
