  For a nonlinear system of coupled PDEs, that describes evolution of a viscous
thin liquid sheet and takes account of surface tension at the free surface, we
show exponential $(H^1,\,L^2)$ asymptotic decay to the flat profile of its
solutions considered with general initial data. Additionally, by transforming
the system to Lagrangian coordinates we show that the minimal thickness of the
sheet stays positive for all times. This result proves the conjecture formally
accepted in the physical literature [1], that a viscous sheet can not rupture
in finite time in absence of external forcing. Moreover, in absence of surface
tension we find a special class of initial data for which the Lagrangian
solution exhibits $L^2$-exponential decay to the flat profile.
