Let $X$ be a Banach space. Evans (page 285) defined the space $L^p(0, T, X)$, then defined the weak time derivative for ${\bf u}\in L^1(0, T, X)$, i.e. one says ${\bf v}\in L^1(0, T, X)$ is the weak derivative of ${\bf u}$, if for any $\phi\in C_0^\infty(0, T)$ we have $$ \int_0^T \phi'(t){\bf u}(t) dt=-\int_0^T \phi(t){\bf v}(t) dt. $$ In this case one writs ${\bf u}'={\bf v}$.
Then Evans went on to discuss a situation when ${\bf u}$ and ${\bf u}'$ are in different spaces. In particular, on page 287 he studied a situation when ${\bf u}\in L^2(0, T, H_0^1(U))$ while ${\bf u}'\in L^2(0, T, H^{-1}(U))$. My question is, in this case, what is the definition of ${\bf u}'$? So far the old definition says ${\bf u}, {\bf u}'$ are supposed to be both in some $L^1(0, T, X)$...
It seems to be like this: ${\bf u}\in L^2(0, T, H_0^1(U))$, so ${\bf u}\in L^1(0, T, H_0^1(U))$. The old definition says that ${\bf u}'$, if exists, is an element in $L^1(0, T, H_0^1(U))$; however, it may not be in $L^2(0, T, H_0^1(U))$.
But then we have the inclusion $H_0^1(U)\subset L^2(U)\subset H^{-1}(U)$, and indeed we have for any $w\in H_0^1(U)$ $$ \|w\|_{H^{-1}(U)}\leq \frac{1}{\sqrt{1+\lambda}}\|w\|_2\leq \frac{1}{1+\lambda}\|w\|_{H_0^1(U)}. $$ So it is still possible ${\bf u}'\in L^2(0, T, H^{-1}(U))$. I guess that is assumed in Evans.