  We study theoretically the profile evolution of a thin viscoelastic film
supported onto a no-slip flat substrate. Due to the nonconstant initial
curvature at the free surface, there is a flow driven by Laplace pressure and
mediated by viscoelasticity. In the framework of lubrication theory, we derive
a thin film equation that contains local viscoelastic stress through the
Maxwell model. Then, considering a sufficiently regular small perturbation of
the free surface, we linearise the equation and derive its general solution. We
analyse and discuss in details the behaviour of this function. We then use it
to study the viscoelastic evolution of a Gaussian initial perturbation through
its transient levelling exponent. For initial widths of the profile that are
smaller than a characteristic length scale involving both the film thickness
and the elastocapillary length, this exponent is shown to reach anomalously
high values at the elastic-to-viscous transition. This prediction should in
particular be observed at sufficiently short times in experiments on thin
polymer films.
