  Stochastic conservation laws are often challenging when it comes to proving
existence of non-negative solutions. In a recent work by J. Fischer and G.
Gr\"un (2018, Existence of positive solutions to stochastic thin-film
equations, SIAM J. Math. Anal.), existence of positive martingale solutions to
a conservative stochastic thin-film equation is established in the case of
quadratic mobility. In this work, we focus on a larger class of mobilities
(including the linear one) for the thin-film model. In order to do so, we need
to introduce nonlinear source potentials, thus obtaining a non-conservative
version of the thin-film equation. For this model, we assume the existence of a
sufficiently regular local solution (i.e., defined up to a stopping time
$\tau$) and, by providing suitable conditions on the source potentials and the
noise, we prove that such solution can be extended up to any $T>0$ and that it
is positive with probability one. A thorough comparison with the aforementioned
reference work is provided.
