Consider $W(0,T) = \{u \in L^2(0,T,V) \text{ with }u_t \in L^2(0,T,V^*)\}$. Pick as $V$ any Hilbert space $H^1_0\subseteq V \subseteq H^1$. One can therefore write the integral $\int_\Omega u_t(t)v(t)dx$ for $ u, v \in W(0,T)$. This is to be intended in the distributional way, i.e. $<u_t,v(t)>_{V^*,V}$.
Now, one can surely define, for an invertible mapping $\phi=\phi(x)$ (with $\phi, \phi^{-1} \in W^{1,\infty}$), the function $u_\phi=u_\phi(t)=u(t)\circ \phi$ on $\Omega_\phi=\phi^{-1}(\Omega)$.
Is $u_\phi$ still $W(0,T)$ and in what sense can one write the change of variables $\int_\Omega u_t(t)v(t)dx = \int_{\Omega'}|\det J\phi|(u_\phi)_t(t)v_\phi(t)dx$ true?
I was thinking of writing the pairing with the derivative, in terms of Riesz representatives. The problem is that gradients in space then appear, because the scalar product with respect to which one is taking representatives is the $H^1$ one. This doesn't allow a nice factoring. Do you have any idea? Is the distributional multiplication intended here?