Consider a thin fluid film whose bottom is rigid and the top free surface contains a monolayer of a surfactant. The hyperbolic-parabolic equation describing the mass conservation for the fluid and the surfactant, derived by M. S. Borgas and J. B. Grotberg [J. Fluid Mech. 193, 151-170 (1988; Zbl 0643.76032)], forms the basis of the paper. Existence of solutions locally in time for given initial data is investigated in detail. It is further shown that the smooth solutions generally do not exist globally in time, because the hyperbolic part of the equation leads to development of shocks. Finally, the author etablishes the stability of solution in a bounded domain with smooth boundary subject to the homogeneous Neumann boundary condition.