I am looking for a higher regularity result for the solution of the problem
$$\partial_t u+div(-A(t,x)\nabla u)=f$$
in a bounded smooth domain $\Omega$ with homogeneous Dirichlet boundary condition. What should be the minimum regularity I should assume for $A$, $f$, and the initial condition such that my solution lies in some $H^k$ space with $k>2$? Also, I would like to know the energy estimates for the $H^k$ norm.
Note: Evans PDE book discusses the case when $A$ is a constant matrix but I am looking for situation where $A$ is both space and time-dependent, bounded, and elliptic.