Source-type solutions to thin-film equations in higher dimensions

Abstract

We prove that the thin film equation h t +div ( h n grad (Δ h ))=0 in dimension d [ges ]2 has a unique C 1 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< n 3 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]).