The ``linear'' limit of thin film flows as an obstacle-type free boundary problem (Q2727765)

This is an excellent paper. The authors study the case of the limit when \(n\) tends to zero of the nonnegative, selfsimilar source-type solutions of the thin film equationsNEWLINE\[NEWLINEu_t+(u^nu_{xxx})_x=0.NEWLINE\]NEWLINE They obtain a unique limiting function \(u\), which is a solution of an obstacle-type free boundary value problem, with the constraint \(u\) larger or equal to zero, associated with the linear equation of the form NEWLINE\[NEWLINEu_t+u_{xxxx}=0.NEWLINE\]NEWLINE 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. Moreover, the authors give an integral representation of \(u\) in the positivity set. They set up a precise definition of the general (non-selfsimiliar) obstacle-type free boundary value problem, which is different from a standard parabolic variational inequality, and compare it with the Cauchy problem. They 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 equation \(u_t=(u^nu_x)_x\) in the fast diffusion range for all \(n\) strictly negative. For these second-order equations the current problems have globally positive solutions. Hence, they have infinite speed of propagation and the condition \(u\) larger or equal to zero does not generate obstacle problems.