Now we know that $C_c^\infty(\Omega) \subset H^1(\Omega)$ is NOT dense.
We also know that (eg. from Lions' and Magenes) that if $V \subset H \subset V'$ is Hilbert triple, $$\mathcal{D}(0,T;V) \subset W(0,T)\quad\text{ is dense},$$ where $\mathcal{D}(0,T;V)$ is the space of infinitely-differentiable compactly supported functions with values in $V$ and $W(0,T) = \{ u \in L^2(0,T;V) : u' \in L^2(0,T;V')\}$
If we let $V=H=\mathbb{R}$ and $\Omega=(0,T)$, then doesn't this latter result contradict our first result??