Unsteady two-dimensional problem of a thin liquid layer with prescribed time-dependent influx into the layer, position of the influx section, and the thickness of the liquid at this section is studied by methods of asymptotic analysis. The ratio of the rate of the liquid thickness variation at the influx section to the influx velocity plays a role of a small parameter of the problem. The influx parameters are such that the flow in the thin layer is inertia dominated, with gravity, surface tension, and liquid viscosity being approximately negligible. Such flows were studied with respect to several applications, some of which are listed in the Introduction. One of the applications concerns with splashing during droplet impact onto a rigid substrate and related kinematic discontinuity propagating along the spray sheet, which is produced by the spreading droplet. This type of splashing was studied by Yarin and Weiss [“Impact of drops on solid surfaces: Self-similar capillary waves, and splashing as a new type of kinematic discontinuity,” J. Fluid Mech. 283, 141–173 (1995)] within a quasi-one-dimensional approach averaging the flow velocity over the layer thickness. We also start with the one-dimensional thin-layer approximation assuming the influx flow is accelerated. Such influx conditions lead to unbounded growth of the thickness of the liquid layer at a certain location and at a certain time instant within the one-dimensional approach. The present study recovers for the first time the structure of the flow close to this singularity using methods of asymptotic analysis. To this aim, the second-order outer solution, which is valid outside the region of the unbounded flow, is derived. The second-order outer solution is used to find proper stretched inner variables and the equations governing the inner flow at the leading order. It is shown that the inner free-surface flow in the stretched variables is two-dimensional, potential, non-linear, and independent of any parameters of the original problem.