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We study the long-time behaviour of solutions to quasilinear doublydegenerate parabolic problems of fourth order. The equations model for instancethe dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeablebottom and with zero contact angle. We consider a shear-rate dependent fluidthe rheology of which is described by a constitutive power-law or Ellis-law forthe fluid viscosity. In all three cases, positive constants (i.e. positive flatfilms) are the only positive steady-state solutions. Moreover, we can give adetailed picture of the long-time behaviour of solutions with respect to the$H^1(\Omega)$-norm. In the case of shear-thickening power-law fluids, oneobserves that solutions which are initially close to a steady state, convergeto equilibrium in finite time. In the shear-thinning power-law case, we findthat steady states are polynomially stable in the sense that, as time tends toinfinity, solutions which are initially close to a steady state, converge toequilibrium at rate $1/t^{1/\beta}$ for some $\beta > 0$. Finally, in the caseof an Ellis-fluid, steady states are exponentially stable in $H^1(\Omega)$.

Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order