Qualitative studies are carried out of quasi-linear parabolic equation in the domain with free mobile boundary. The movement of a boundary is described by integral cinematic boundary condition. The existence and uniqueness is proved of the local in time solution of corresponding one-dimensional boundary problem in Holder spaces. The researched boundary problem is a mathematical model of the consolidation of saturated porous media.

quasi-linear parabolic equations, free boundary, existence and uniqueness of solution.