THE FREE-BOUNDARY OF THIN VISCOUS FLOWS

We consider the fourth order nonlinear degenerate parabolic equation u (t) + (u(n)u(xxx))(x) = 0 which arises in lubrication models for thin viscous films and spreading droplets as well as in the flow of a thin neck of fluid in a Hele-Shaw cell. Fourth order parabolic equations wi th a similar type of degeneracy and/or additional fouler order terms a rise in the theory of phase separation, for binary alloys (Cahn-Hillia rd equation with degenerate mobility), in some plasticity models and i n the study of spatial pattern formation in biological systems. This a uthor has recently proved that if 0 < n < 2 the above equation has fin ite speed of propagation for nonnegative strong solutions and hence th ere exists an interface or free boundary separating the regions where u > 0 and u = 0. Furthermore, the interface is Holder continuous if 1/ 2 < n < 2 and right-continuous if 0 < n less than or equal to 1/2. Fin ally we consider the Cauchy problem and present the recent results on optimal asymptotic rates as t --> infinity for the solution and for th e interface when 0 < n < 2; these rates exactly match those of the sou rce-type (fundamental) solutions.