Mathematical modelling of the initial attachment process in biofilm formation is analyzed qualitatively. We consider the Wanner and Guyer model and discuss the free boundary value problem for the system of nonlinear hyperbolic partial differential equations and ordinary equations arising when the initial biofilm thickness is zero. The main difficulty of the mathematical problem is related to the space-like free boundary. This aspect makes the problem completely different from the usual free boundary problems describing biofilm growth where the free boundary is a time-like line. In the present case the initial data are assigned on the free boundary and depend on the attachment flux as well as the bacterial concentrations in the bulk liquid. The original coordinates are transformed to characteristic coordinates and the differential equations are converted to integral equations. The qualitative analysis provides insights into uniqueness, existence and properties of solutions. Some numerical simulations are provided. Generalizations to the mathematical problem are also discussed, e.g. the initial phase of the biofilm reactor model.