  We study the dynamic behaviour of solutions to a fourth-order quasilinear
degenerate parabolic equation for large times arising in fluid dynamical
applications. The degeneracy occurs both with respect to the unknown and with
respect to the sum of its first and third spatial derivative. The modelling
equation appears as a thin-film limit for the interface separating two
immiscible viscous fluid films confined between two cylinders rotating at small
relative angular velocity. More precisely, the fluid occupying the layer next
to the outer cylinder is considered to be Newtonian, i.e. it has constant
viscosity, while we assume that the layer next to the inner cylinder is filled
by a shear-thinning power-law fluid.
  Using energy methods, Fourier analysis and suitable regularity estimates for
higher-order parabolic equations, we prove global existence of positive weak
solutions in the case of low initial energy. Moreover, these global solutions
are polynomially stable, in the sense that interfaces which are initially close
to a circle, converge at rate $1/t^{1/\beta}$ for some $\beta > 0$ to a circle,
as time tends to infinity.
  In addition, we provide regularity estimates for general nonlinear degenerate
parabolic equations of fourth order.
