Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and $\mathbb{R} \ni T > 0$. I will abbreviate $X=H^1(\Omega)$ and write $X'$ for its topological dual.
Given $$u\in L^2\left(0,T;X \right)$$ with weak (i.e.: distributional) time derivative $$u_t \in L^2\left( 0,T;X' \right)$$ and some function $$f \in C^1\left( \mathbb{R} \right)$$ satisfying (at least) $$f\circ u(t) \in L^1(\Omega) \quad \textrm{for. a.e. } t\in (0,T).$$
I define $$ F\colon (0,T) \rightarrow \mathbb{R}, \quad t \mapsto \int\limits_\Omega f\circ u(t) \, dx.$$
Now, the question is: Which further conditions need to be imposed on $f$ to make $F$ weakly differentiable with $F_t \in L^1((0,T))$ and (probably)
$$ F_t = {\left\langle u_t(\cdot), f' \circ u(\cdot) \right\rangle}_{X',X} $$
which would be some kind of chain rule.
Obviously, it is necessary for $F'\circ u(t)$ to be in $X$ for a.e. $t\in (0,1)$. But how can that be guaranteed? Does anybody have a good book as a reference for this type of problem?
PS: I came across this similar question, but did not find the answer precise enough in terms of what requirements $f$ needs to meet.