The "Linear" Limit of Thin Film Flows as an Obstacle-Type Free Boundary Problem

We study the limit as $n\to 0$ of the nonnegative, self-similar source-type solutions of the thin film equationu _t + (u ⁿ u _xxx ) _x =0. We obtain a unique limiting function u, which is a solution of an obstacle-type free boundary problem, with the constraint $u\geq 0$, associated with the linear equationu _t + u _xxxx =0. The function u is C ¹ for t > 0 and has finite speed of propagation, the positivity set {uM>0} being bounded by two contact lines $x=\pm a t^{1/4}$ (a constant). The function u has a Dirac mass as initial condition and satisfies the linear equation in the positivity set, but not across the free boundaries or contact lines, also known as moving boundaries. We give an integral representation of u in the positivity set. We set up a precise definition of the general (non-self-similar) obstacle-type free boundary problem, which is different from a standard parabolic variational inequality, and compare it with the Cauchy problem. We also consider source-type solutions for negative values of n, which are solutions of the obstacle-type free boundary problem (rather than the Cauchy problem) and still have finite speed of propagation. The situation is rather different from that of the heat equation u _t =u _xx and the porous media equationu _t =(u ⁿ u _x ) _x in the fast diffusion rangen < 0. For these second-order equations the current problems have globally positive solutions. Hence, they have infinite speed of propagation and the condition $u\geq 0$ does not generate obstacle problems.