  We study the steady states and dynamics of a thin film-type equation with
non-conserved mass in one dimension. The evolution equation is a nonlinear
fourth-order degenerate parabolic PDE motivated by a model of volatile viscous
fluid films allowing for condensation or evaporation. We show that by changing
the sign of the non-conserved flux and breaking from a gradient flow structure,
the problem can exhibit novel behaviors including having two distinct classes
of coexisting steady state solutions. Detailed analysis of the bifurcation
structure for these steady states and their stability reveal several
possibilities for the dynamics. For some parameter regimes, solutions can lead
to finite-time rupture singularities. Interestingly, we also show that a finite
amplitude limit cycle can occur as a singular perturbation in the
nearly-conserved limit.
