uniqueness for non homogenous heat equation

Question

hi in nonhomogenous heat equation :$u_{t}-\Delta u=f(x,t)$ in U smooth open of $R^{n}$ and $f \in L^{2}((0,T),L^{2}(U))$ with $u_{0}$ in $L^{2}(U)$,$u=0$ in $\partial U$ i find that u $\in C((0,T),L^{2}(U)) \cap L^{2}((0,T),H_{0}^{1}(U))$ and i wanted to prove uniqueness but in the hint thay say that $u_{t}\in L^{2}((0,T),H_{0}^{1}(U)\cap H^{2}(U))$ i can prove that but my question is why it's a a condition to prove uniqueness? thanks