SOURCE-TYPE SOLUTIONS TO THIN-FILM EQUATIONS IN HIGHER DIMENSIONS

We prove that the thin film equation h(t) + div (h(n) grad(Delta h)) = 0 in dimension d greater than or equal to 2 has a unique C-1 source-t ype radial self-similar non-negative solution if 0 < n < 3 and has no solution of this type if n greater than or equal to 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 sep arating the regions where h > 0 and h = 0. (The case d = 1 was already known [1]).