I'm a bit confused about how Evans refers to derivatives at some points and if he means weak derivative. In particular on page 301 he gives the definition that if $\textbf{u} \in L^1(0,T;X)$ and $\textbf{v} \in L^1(0,T;X)$, then we say that $\textbf{v}$ is the weak derivative of $\textbf{u}$ (written $\textbf{u'}$ = $\textbf{v}$) if:
$\int_0^T \phi'(t) \textbf{u}(t) dt = - \int_0^T \phi(t) \textbf{v}(t) dt$
for all test functions $\phi \in C_c^\infty (O,T)$. Now if we skip to chapter 7, on the first time he introduces a bold u with a ' (ie $\textbf{u'}$) on the same line in parenthesis it has ($' = \frac{d}{dt}$). This is on page 374 of Evans. So are we not refering to ' in the sense of a weak derivative above?. Part of my concern is that on the same page he discusses having $\textbf{u}\in L^2(0,T;H_0^1(U))$ and $\textbf{u}' \in L^2(0,T;H^{-1}(U))$. But there seems to be some details that seem odd to be in terms of the definition of weak derivative given above. i.e., above $\textbf{u}$ and $\textbf{v}$ are given to be elements of the same space, but I thought $H_0^1 \subset H^{-1}$ and not equal so this seems to not fit with the X being the same for $\textbf{u}$ and $\textbf{v}$ in the definition of weak derivative.
Overall, I suspect that I simply have a lack of understanding of the basic fundamentals, but I don't know where to start.
EDIT I'm using Partial Differential Equations 2cnd ed