  We consider the anisotropic mean curvature flow of entire Lipschitz graphs.
We prove existence and uniqueness of expanding self-similar solutions which are
asymptotic to a prescribed cone, and we characterize the long time behavior of
solutions, after suitable rescaling, when the initial datum is a sublinear
perturbation of a cone. In the case of regular anisotropies, we prove the
stability of self-similar solutions asymptotic to strictly mean convex cones,
with respect to perturbations vanishing at infinity. We also show the stability
of hyperplanes, with a proof which is novel also for the isotropic mean
curvature flow.
