We study a boundary value problem for a singular parabolic minimal surface equation that originates from the elliptic "hanging roof problem". By applying a fixed point argument it is proven, that a unique classical solution exists, if the boundary and initial values are chosen "large enough" and the domain has non-negative inward mean curvature, improving earlier results, which required strictly positive inward mean curvature. The solution is smooth in the domain's interior, continuous up to the boundary and converges to a solution of the corresponding stationary (i.e. time independent) problem, if the domain's boundary is smooth enough. Furthermore it is shown that in case of "too small" boundary and initial values, the solution remains smooth until it reaches the value zero in finite time, where it becomes irregular. Under additional assumptions it is proven that at the time at which the solution reaches the value zero, it remains Hölder continuous.