  In this paper, we extend the results of [1] by proving exponential asymptotic
$H^1$-convergence of solutions to a one-dimensional singular heat equation with
$L^2$-source term that describe evolution of viscous thin liquid sheets while
considered in the Lagrange coordinates. Furthermore, we extend this asymptotic
convergence result to the case of a time inhomogeneous source. This study has
also independent interest for the porous medium equation theory.
