Consider the parabolic problem $$\left\{\hspace{5pt}\begin{aligned} &-\dfrac{\partial u }{\partial t} +a_{ij}\partial_{ij}u +b_i \partial_i u +c u = f(u) & \hspace {10pt} &\text{for $(x,t) \in \Omega \times (0,T]$} ;\\ &u(x,t) = g(x,t) & \hspace{10pt} &\text{for $(x,t) \in \partial_p (\Omega \times (0,T]) $.} \end{aligned}\right.$$ $\partial_p$ denotes the parabolic boundary. Here we assume that $u$, $a_{ij}>0$, $b$ and $c$ are $C^{\alpha,\alpha/2}(\Omega \times [0,T])$, while $f \in C^{2+ \alpha}(\Omega)$ and $g \in W^{2,1}_2(\Omega \times [0,T]) \cap C^{\alpha,\alpha/2}(\Omega \times [0,T])$ for some $\alpha$.
Now I have the existence of strong solution $u \in W^{2,1}_2(\Omega \times [0,T]) \cap C^{\alpha,\alpha/2}(\Omega \times [0,T])$. How can I prove that it is actually a classical solution?
I am going to prove that $u$ has a higher regularity, say $W^{2k+2,k+1}_2$ for some $k$, then I can use embedding theorem to show that $u \in C^{2+\alpha,1+\alpha/2}\in (\Omega \times (0,T])$ is classical. But I can not find any reference for higher regularity. Can I have a book or text for it? Or may be there is another way to establish this?