Regularity of Parabolic pde In Evans' pde Book, In Theorem 5, p. 360 (old edition) which concern regularity of parabolic pdes. he consider the case where the coefficients $a_{ij},b_i,c$ of the uniformly parabolic operator (divergent form) $L$ coefficients are all smooth and don't depend on the time parameter $t$
\begin{cases}
\mathbf{u}_t + L \mathbf{u}= \mathbf{f} &\text{ in } U\times [0,T] \\
\mathbf{u} = 0 &\text{ in } \partial U \times [0,T]\\
\mathbf{u}(0) = g &\text{ in } U
\end{cases}
where $\mathbf{f} \in L^{2}(0,T; L^2(U))$, $g \in H^1_0(U)$ and $U$ is a open, bounded subset of $\mathbb{R}^N$ with smooth boudary.
The second assertion (ii) of theorm is the following
if in addition $g \in H^1_0(U)\cap H^2(U)$ and $\mathbf{f} \in H^{1}(0,T; L^2(U))$ then we have
(ii) $u' \in L^\infty(0,T; L^2(U) \cap L^2(0,T; H_1^0(U))$ and $u'' \in L^\infty(0,T; H^{-1}(U)$.
For proving (ii) Evans use Galerkin approxitamtion.
My question is, is there any generalization for proving regularty of time deivative u' in general abstract Cauchy problem, or can we obtain, at least, the previous result by semigroups theory ? Thank you for any help.
 A: Yes, there is a generalization of this result. Let us again consider the abstract Cauchy problem
$$\begin{cases}
u_t + L u= f &\text{ in } U\times [0,T] \\
u(0) = g &\text{ in } U
\end{cases}$$
equipped with some boundary conditions. We consider the Gelfand tripel $V \hookrightarrow H \hookrightarrow V'$. In your case we had $V=H_0^1(U)\cap H^2(U), H=L^2(U)$ with homogeneous Dirichlet boundary. This is the first generalization.
First of all, the general existence result gives us a unique solution $$u \in L^2(0,T;V) \cap H^1(0,T;V')$$
if we assumed $f\in L^2(0,T;V'), g \in H$. See e.g. Wloka's book "Partielle Differentialgleichungen".
Now, you are interested in regularity results. And yes, we can generalize your result. For the solution $u$ we get the regularity
$$u \in H^k(0,T;V) \cap H^{k+1}(0,T;V')$$
if we assumed $f \in  H^k(0,T;V')$, $\frac{d^j u}{dt^j}(0) \in V$ for all $j=0,...,k-1$ and $\frac{d^k y}{dt^k}(0)\in H$.
You can also check that for $\mathbf{k=0}$ the existence result generalizes to the one above and for $\bf{k=1}$ we have
$$u \in H^1(0,T;V) \cap H^2(0,T;V')$$
if we assumed $f \in H^1(0,T;V'), g \in V, u_t(0) \in H$. This is what Evans has. Indeed, he additionally assumed $f\in H^1(0,T;H) \hookrightarrow C([0,T];H)$ which implies $u_t(0) \in H$. Further, the $L^\infty$ spaces can be achieved from several embedding results, e.g. $$u' \in L^2(0,T;V) \cap H^1(0,T;V') \hookrightarrow C([0,T];H).$$
