Consider the second order linear parabolic PDE: \begin{eqnarray*} \partial_{t}u + Lu &= f && \text{ in $\Omega \times (0,T]$},\\ u &= 0 && \text{ on $\partial \Omega \times [0,T]$}, \\ u & = g && \text { on $\Omega \times \{t =0\}$}; \end{eqnarray*} where $f,g$ and the coefficients of the general second order linear elliptic operator $L$ are smooth in space and time; moreover $\Omega$ is a smooth domain.
I am interested in $H^2$-regularity of $u$. In the book of Evans (2nd edition), Theorem 5, Chapter 7, it is proved by Galerkin approximations that $u \in L^2(0,T; H^2(\Omega))$ whenever the coefficients of $L$ are independent of time.
Does the same regularity result holds for the case when the coefficients of $L$ are functions of space as well as time? Also, does the same regulariy result holds in the time dependent case if we replace the homogeneous Dirichlet boundary condition by the homogeneous Neumann boundary condition? Any suitable reference would be helpful.