Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 95, pp. 1-29.

Title: Heteroclinic orbits, mobility parameters and stability 
       for thin film type equations

Authors: Richard. S. Laugesen (Univ.Illinois, Urbana, IL, USA)
         Mary C. Pugh (Univ. of Toronto, ON, Canada)
         
Abstract: 
   We study the phase space of the evolution equation
   $$
   h_t = -(h^n h_{xxx})_x - \mathcal{B} (h^m h_x)_x ,
   $$
   where $h(x,t) \geq 0$. The parameters $n>0$, $m \in \mathbb{R}$, 
   and the Bond number $\mathcal{B}>0$ are given.
   We find numerically, for some ranges of $n$ and $m$, that
   perturbing the positive periodic steady state in a certain direction
   yields a solution that relaxes to the constant steady state. Meanwhile
   perturbing in the opposite direction yields a solution that appears to
   touch down or `rupture' in finite time, apparently approaching a
   compactly supported `droplet' steady state.
   We then investigate the structural stability of the evolution by
   changing the mobility coefficients, $h^n$ and $h^m$.  We find evidence
   that the above heteroclinic orbits between steady states are perturbed
   but not broken, when the mobilities are suitably changed.
   We also investigate touch-down singularities, in which the solution
   changes from being everywhere positive to being zero at isolated
   points in space.  We find that changes in the mobility exponent $n$ 
   can affect the  number of touch-down points per period, 
   and affect whether these singularities occur in finite or infinite time.
   
Submitted February, 28, 2002. Published November 5, 2002
Math Subject Classifications: 35K55, 37C29 ,37L15, 76D08.
Key Words: Nonlinear PDE of parabolic type; heteroclinic orbits;
           stability problems; lubrication theory.