Assume we have a parabolic PDE. For sake of simplicity consider the heat equation defined on a "nice" domain: $$\partial_t u(x,t)-\Delta u(x,t)=0$$ We know that the weak form of this PDE is: $$\left< \dot{u}(t),v \right>_{H^1}+\left( \nabla u(t),\nabla v \right)_{\mathcal{L}^2}=0\qquad \forall \;v\in H^1 \qquad \forall\; t\;\;\mathrm{a.e.}$$ Where $\left< \cdot, \cdot\cdot \right>_{H^1}$ is representing the duality product in $H^1$.
In general, is it true that the derivative $u(t)$ of a function $u\in\mathcal{L}\left( 0,T;H^1 \right)$ is in the dual space ${H^1}^*$?