  The large-time behavior of solutions to the thin film equation with linear
mobility in the complete wetting regime on $\mathbb{R}^N$ is examined: We
investigate the higher order asymptotics of solutions converging towards
self-similar Smyth--Hill solutions under certain symmetry assumptions on the
initial data. The analysis is based on a construction of finite-dimensional
invariant manifolds that solutions approximate to an arbitrarily prescribed
order.
