I am looking at the classical proof of uniqueness for the heat equation in Evans.
Clearly, we differentiate under the integral sign of the square of $w$. A very basic question is, why are the assumptions of $u$, $Du$ and $\Delta u$ in $L^2$ not imposed here? When do we need them to be imposed in this problem?
The $L^2$ assumptions are not imposed because the assumption $u\in C_1^2(\overline{U}_T)$ supersedes them. This is because $U$ is assumed to be bounded and $T>0$ is fixed, so $\overline{U}_T$ is a compact domain. (You can assume $T$ is finite, because in the case $T=\infty$ you can just keep proving uniqueness for successively larger values of $T$.) Since $u$ and its derivatives (both spatial and temporal) are assumed continuous, they must be bounded on $\overline{U}_T$, and a bounded function on a compact domain is necessarily $L^2$.
Therefore $L^2$ conditions do not need to be imposed in this theorem; you already have something stronger. One thing you can do is weaken your assumptions on $u$ and the boundary condition to weak differentiability. There are still energy estimate techniques to prove uniqueness of weak solutions (which work more generally for parabolic PDEs), but you will need to impose $L^2$ conditions on the weak derivatives of $u$ for all the integrals to make sense.