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

This paper is concerned with the one-dimensional thin film equation View the MathML source{∂u∂t+∂∂x(M(u)∂∂x[∂2u∂x2−P(u)])=0P(u)=1un−ϵm−num,00 Turn MathJax on in (0,L)×R+(0,L)×R+ with the homogeneous Neumann boundary conditions View the MathML source(uxx−P(u))x|x=0,L=0,ux|x=0,L=0,for all t>0. Turn MathJax on 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.