Fluid accumulation in thin-film fl ows driven by surface tension and   gravity (I): Rigorous analysis of a drainage equation
  We derive a boundary layer equation describing accumulation regions within a thin-film approximation framework where gravity and surface tension balance. As part of the analysis of this problem we investigate in detail and rigorously the 'drainage' equation (phi"'+1)phi^3=1. In particular, we prove that all solutions that do not tend to 1 as the independent variable goes to infinity are oscillatory, and that they oscillate in a very specific way. This result and the method of proof will be used in the analysis of solutions of the afore mentioned boundary layer problem. 
C. M. Cuesta and J. J. L. Velazquez
math.AP math.DS