On smoothness up to the boundary of solutions of parabolic equations with a degenerate operator

Parabolic equations $\partial_tu=-Au+f_0$, $ u|_{t=0}=f_1$ are considered that are of second order with a nonnegative quadratic form $a(x,\zeta)$ corresponding to the space variables. This form degenerates on the boundary: $a(x,\nu)=0$, where $\nu$ is the normal vectors, which corresponds to the condition of impermeability of the boundary. Special function spaces $E^s$ with weight are introduced. Semiboundedness of the operator $A$ in these spaces with arbitrary $s$ is proved: $(Av,v)_{E^s}\geqslant -C\|v\|_{E^s}^2$. On this basis theorems on the smoothness of solutions for $f_0,f_1\in E^s$ are proved. Theorems on the smoothness of solutions $u(x)$ of the elliptic equation $Au + \lambda u = f_0$ are also obtained.