On the partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth

We consider quasilinear nondiagonal parabolic systems with quadratic growth on the gradient in a parabolic cylinder $Q$. Under Dirichlet and Neumann boundary conditions partial Hölder continuity up to the lateral surface of $Q$ of solutions $u\in W_2^{1,1} (Q)\cap L^\infty(Q)$ is proved. Hausdorff dimension of a singular set is estimated. In the proof we get of the maximum principle theorem for corresponding model linear problems.