Propagation of support in one-dimensional convected thin-film flow

In one space dimension, we study the finite speed of propagation property for zero contact--angle solutions of the thin-film equation in presence of a convective term. In the case of strong slippage, we obtain bounds in terms of the initial mass for both the ``fast" and the ``slow" interfaces, and for both short and (whenever the solution is global) large times, which we expect to be sharp. In the case of weak slippage, we obtain partial results for short times, which include a quantitative bound for moderate growths of the convective term. Our approach is based on energy/entropy methods shaped upon suitable extensions of Stampacchia's Lemma