Let $\Omega(t)$ be a bounded domain for each $t$. Let $Q=\bigcup_{t \in [0,T]} \Omega(t) \times \{t\}$ and $S=\bigcup_{t \in [0,T]} \partial\Omega(t) \times \{t\}$.
The normal vector to $S$ at $(x,t)$ is $\nu = (\nu_x, \nu_t)$.
We have by integration by parts $$\int_{Q}v\varphi_t\;dxdt = -\int_S \varphi v|_{S} \nu_t - \int_Q v_t \varphi\;dxdt$$
I don't understand how the integration in parts is done here. Since we are just integrating over time, which is 1D, why don't we just get a term which is a difference between $t=T$ and $t=0$? What is the integration by parts formula used here?