Counting the stationary states and the convergence to equilibrium for the 1-D thin film equation

This paper is concerned with the one-dimensional thin film equation {∂u∂t+∂∂x(M(u)∂∂x[∂2u∂x2−P(u)])=0P(u)=1un−ϵm−num,00 in (0,L)×R+(0,L)×R+ with the homogeneous Neumann boundary conditions (uxx−P(u))x|x=0,L=0,ux|x=0,L=0,for all t>0. We prove that for any given positive initial datum, the number of positive stationary states is at most infinitely countable. Furthermore, we prove that the solution of the evolution problem converges to an equilibrium as time tends to infinity.