Let us consider the linear parabolic PDE: \begin{equation} \partial_t u - \Delta u=f \text{ and }u(0)=u_0 \end{equation} where $u_0 \in L^2(\mathbb{R})$ and $f \in L^2_t([0,T], L^2_x(\mathbb{R}))$.
Then, the definition of a weak solution is, according to Evans PDE, that \begin{equation} \langle u(t), v \rangle_{L^2_x}+ \langle \nabla u(t), \nabla v \rangle_{L^2_x}=\langle \nabla f(t), \nabla v \rangle_{L^2_x} \end{equation} for each $v \in H^1(\mathbb{R})$ and a.e. $0 \leq t \leq T$ as well as $u(0)=g$
Here, I am extremely confused about the a.e. $0 \leq t \leq T$ condition. That is,
Does the set of $t \in [0,T]$ that satisfies the above equation "depend on" each $v$??
This seems quite tricky and no reference seems to state the exact conditions..
Could anyone please help me?