Solvability quasi-linear parabolic equation in domains with picewise monotone boundary

We investigate the existence of regular solutions for the quasilinear parabolic equation in non-cylindrical domain with boundary of class $W^1_2$. The equation can degenerate to the decision. Approximate solutions are constructed using the projection method of the family of projectors depending on the time parameter. We prove that a limit of these solutions will be the solution of the problem. To justify the existence of the limit methods are used for the functions of the compact scale of Banach spaces.