Regularity of solutions of the model Venttsel' problem for quasilinear parabolic systems with nonsmooth in time principal matrices

The Venttsel' problem in the model statement for quasilinear parabolic systems of equations with nondiagonal principal matrices is considered. It is only assumed that the principal matrices and the boundary condition are bounded with respect to the time variable. The partial smoothness of the weak solutions (Hölder continuity on a set of full measure up to the surface on which the Venttsel' condition is defined) is proved. The proof uses the $A(t)$-caloric approximation method, which was also used in [1] to investigate the regularity of the solution to the corresponding linear problem.